Content check — Calculus Vol. 3, Chapter 2: Vectors in Space

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 2: Vectors in Space

270 problems · 81 figures

Checkpoint 2.1 §2.1 source ↗

Sketch the vector

\vec{S T}

where

S

is point

(3, −1)

and

T

is point

(−2, 3) .

Answer

This figure is a graph of the coordinate system. There is a line segment beginning at the ordered pair (3, -1). Also, this point is labeled “S.” The line segment ends at the ordered pair (-2, 3) and is labeled “T.” There is an arrowhead at point “T,” representing a vector. The line segment is labeled “ST.”

Checkpoint 2.2 §2.1 source ↗

Using vectors

v

and

w

from Example 2.2, sketch the vector

2 w - v .

Answer

This figure is a triangle formed by having vector 2w on one side and vector -v adjacent to 2w. The terminal point of 2w is the initial point of -v. The third side is labeled “2w – v.”

Checkpoint 2.3 §2.1 source ↗

Which of the following vectors are equivalent?

Answer

Vectors

a,

b,

and

e

are equivalent.

Checkpoint 2.4 §2.1 source ↗

Vector

w

has initial point

(−4, −5)

and terminal point

(−1, 2) .

Express

w

in component form.

Answer

〈 3, 7 〉

Checkpoint 2.5 §2.1 source ↗

Let $a = 〈 7, 1 〉$ and let $b$ be the vector with initial point $(3, 2)$ and terminal point $(−1, −1) .$

a. Find $‖ a ‖ .$

b. Express $b$ in component form.

c. Find $3 a - 4 b .$

Answer

‖ a ‖ = 5 \sqrt{2},

b = 〈 −4, −3 〉,

3 a -

4 b =

〈 37,

15 〉

Checkpoint 2.7 §2.1 source ↗

Find the component form of vector

v

with magnitude

10

that forms an angle of

120 °

with the positive x-axis.

Answer

v = 〈 −5, 5 \sqrt{3} 〉

Checkpoint 2.8 §2.1 source ↗

Let

v = 〈 9, 2 〉 .

Find a vector with magnitude

5

in the opposite direction as

v .

Answer

〈 -

\frac{45}{\sqrt{85}},

-

\frac{10}{\sqrt{85}} 〉

Checkpoint 2.9 §2.1 source ↗

Let

a = 〈 16, −11 〉

and let

b

be a unit vector that forms an angle of

225 °

with the positive x-axis. Express

a

and

b

in terms of the standard unit vectors.

Answer

a = 16 i - 11 j,

b = - \frac{\sqrt{2}}{2} i - \frac{\sqrt{2}}{2} j

Checkpoint 2.10 §2.1 source ↗

An airplane flies due north at an airspeed of

550

mph. The wind is blowing from the northwest at

50

mph. What is the ground speed of the airplane?

Answer

Approximately

516

mph

2-1 §2.1 source ↗

For the following exercises, consider points $P (−1, 3),$ $Q (1, 5),$ and $R (−3, 7) .$ Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.

$\vec{P Q}$

Answer

\vec{P Q} = 〈 2, 2 〉;

\vec{P Q} = 2 i + 2 j

2-3 §2.1 source ↗

$\vec{Q P}$

Answer

\vec{Q P} =

〈 −2,

−2 〉;

\vec{Q P} = −2 i - 2 j

2-5 §2.1 source ↗

$\vec{P Q} + \vec{P R}$

Answer

\vec{P Q} +

\vec{P R} =

〈 0,

6 〉;

\vec{P Q} + \vec{P R} = 6 j

2-7 §2.1 source ↗

$2 \vec{P Q} - 2 \vec{P R}$

Answer

2 \vec{P Q} -

2 \vec{P R} =

〈 8,

−4 〉;

2 \vec{P Q} -

2 \vec{P R} =

8 i -

4 j

2-9 §2.1 source ↗

The unit vector in the direction of $\vec{P Q}$

Answer

〈 \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} 〉;

\frac{1}{\sqrt{2}} i + \frac{1}{\sqrt{2}} j

2-11 §2.1 source ↗

A vector $v$ has initial point $(−1, −3)$ and terminal point $(2, 1) .$ Find the unit vector in the direction of $v .$ Express the answer in component form.

Answer

〈 \frac{3}{5}, \frac{4}{5} 〉

2-13 §2.1 source ↗

The vector $v$ has initial point $P (1, 0)$ and terminal point $Q$ that is on the y-axis and above the initial point. Find the coordinates of terminal point $Q$ such that the magnitude of the vector $v$ is $\sqrt{5} .$

Answer

Q (0, 2)

2-15 §2.1 source ↗

For the following exercises, use the given vectors $a$ and $b .$

$a = 2 i + j,$ $b = i + 3 j$

Answer

a + b = 3 i + 4 j,

a + b = 〈 3, 4 〉;

a - b = i - 2 j,

a - b = 〈 1, −2 〉;

c. Answers will vary; d.

2 a = 4 i + 2 j,

2 a = 〈 4, 2 〉,

− b = − i - 3 j,

− b = 〈 −1, −3 〉,

2 a - b = 3 i - j,

2 a - b = 〈 3, −1 〉

2-17 §2.1 source ↗

For the following exercises, use the given vectors $a$ and $b .$

Let $a$ be a standard-position vector with terminal point $(−2, −4) .$ Let $b$ be a vector with initial point $(1, 2)$ and terminal point $(−1, 4) .$ Find the magnitude of vector $−3 a + b - 4 i + j .$

Answer

15

2-19 §2.1 source ↗

For the following exercises, use the given vectors $a$ and $b .$

Let $u$ and $v$ be two nonzero vectors that are nonequivalent. Consider the vectors $a = 4 u + 5 v$ and $b = u + 2 v$ defined in terms of $u$ and $v .$ Find the scalar $λ$ such that vectors $a + λ b$ and $u - v$ are equivalent.

Answer

λ = −3

2-21 §2.1 source ↗

For the following exercises, use the given vectors $a$ and $b .$

Consider the vector $a (t) =$ $〈 cos t,$ $sin t 〉$ with components that depend on a real number $t .$ As the number $t$ varies, the components of $a (t)$ change as well, depending on the functions that define them.

a. Write the vectors $a (0)$ and $a (π)$ in component form.

b. Show that the magnitude $‖ a (t) ‖$ of vector $a (t)$ remains constant for any real number $t .$

c. As $t$ varies, show that the terminal point of vector $a (t)$ describes a circle centered at the origin of radius $1 .$

Answer

a (0) = 〈 1, 0 〉,

a (π) =

〈 −1,

0 〉;

b. Answers may vary; c. Answers may vary

2-23 §2.1 source ↗

For the following exercises, use the given vectors $a$ and $b .$

Show that vectors $a (t) =$ $〈 cos t,$ $sin t 〉$ and $a (x) =$ $〈 x,$ $\sqrt{1 - x^{2}} 〉$ are equivalent for $x = 1$ and $t = 2 k π,$ where $k$ is an integer.

Answer

Answers may vary

2-25 §2.1 source ↗

For the following exercises, find vector $v$ with the given magnitude and in the same direction as vector $u .$

$‖ v ‖ =$ $7,$ $u =$ $〈 3,$ $4 〉$

Answer

v = 〈 \frac{21}{5}, \frac{28}{5} 〉

2-27 §2.1 source ↗

For the following exercises, find vector $v$ with the given magnitude and in the same direction as vector $u .$

$‖ v ‖ =$ $7,$ $u =$ $〈 3,$ $−5 〉$

Answer

v =

〈 \frac{21 \sqrt{34}}{34},

-

\frac{35 \sqrt{34}}{34} 〉

2-29 §2.1 source ↗

For the following exercises, find the component form of vector $u,$ given its magnitude and the angle the vector makes with the positive x-axis. Give exact answers when possible.

$‖ u ‖ = 2,$ $θ = 30 °$

Answer

u = 〈 \sqrt{3}, 1 〉

2-31 §2.1 source ↗

For the following exercises, find the component form of vector $u,$ given its magnitude and the angle the vector makes with the positive x-axis. Give exact answers when possible.

$‖ u ‖ = 5,$ $θ = \frac{π}{2}$

Answer

u = 〈 0, 5 〉

2-33 §2.1 source ↗

For the following exercises, find the component form of vector $u,$ given its magnitude and the angle the vector makes with the positive x-axis. Give exact answers when possible.

$‖ u ‖ = 10,$ $θ = \frac{5 π}{6}$

Answer

u = 〈 −5 \sqrt{3}, 5 〉

2-35 §2.1 source ↗

For the following exercises, vector $u$ is given. Find the angle $θ \in [0, 2 π)$ that vector $u$ makes with the positive direction of the x-axis, in a counter-clockwise direction.

$u = 5 \sqrt{2} i - 5 \sqrt{2} j$

Answer

θ = \frac{7 π}{4}

2-37 §2.1 source ↗

For the following exercises, vector $u$ is given. Find the angle $θ \in [0, 2 π)$ that vector $u$ makes with the positive direction of the x-axis, in a counter-clockwise direction.

Let $a = 〈 a_{1}, a_{2} 〉,$ $b = 〈 b_{1}, b_{2} 〉,$ and $c = 〈 c_{1}, c_{2} 〉$ be three nonzero vectors. If $a_{1} b_{2} -$ $a_{2} b_{1} \neq 0,$ then show there are two scalars, $α$ and $β,$ such that $c = α a + β b .$

Answer

Answers may vary

2-39 §2.1 source ↗

For the following exercises, vector $u$ is given. Find the angle $θ \in [0, 2 π)$ that vector $u$ makes with the positive direction of the x-axis, in a counter-clockwise direction.

Let $P (x_{0}, f (x_{0}))$ be a fixed point on the graph of the differentiable function $f$ with a domain that is the set of real numbers.

a. Determine the real number $z_{0}$ such that point $Q (x_{0} + 1, z_{0})$ is situated on the line tangent to the graph of $f$ at point $P .$

b. Determine the unit vector $u$ with initial point $P$ in the direction of vector $P Q .$

Answer

z_{0} =

f (x_{0}) +

f^{'} (x_{0});

u =

\frac{1}{\sqrt{1 + {[f^{'} (x_{0})]}^{2}}} 〈 1,

f^{'} (x_{0}) 〉

2-43 §2.1 source ↗

For the following exercises, vector $u$ is given. Find the angle $θ \in [0, 2 π)$ that vector $u$ makes with the positive direction of the x-axis, in a counter-clockwise direction.

Calculate the coordinates of point $D$ such that $A B C D$ is a parallelogram, with $A (1, 1),$ $B (2, 4),$ and $C (7, 4) .$

Answer

D (6, 1)

2-45 §2.1 source ↗

For the following exercises, vector $u$ is given. Find the angle $θ \in [0, 2 π)$ that vector $u$ makes with the positive direction of the x-axis, in a counter-clockwise direction.

The speed of an object is the magnitude of its related velocity vector. A football thrown by a quarterback has an initial speed of $70$ mph and an angle of elevation of $30 ° .$ Determine the velocity vector in mph and express it in component form. (Round to two decimal places.)

Answer

〈 60.62, 35 〉

2-47 §2.1 source ↗

For the following exercises, vector $u$ is given. Find the angle $θ \in [0, 2 π)$ that vector $u$ makes with the positive direction of the x-axis, in a counter-clockwise direction.

A bullet is fired with an initial velocity of $1500$ ft/sec at an angle of $60 °$ with the horizontal. Find the horizontal and vertical components of the velocity vector of the bullet. (Round to two decimal places.)

This figure is the first quadrant of a coordinate system. There is a vector from the origin that is labeled “v sub 0 = 1500 feet per second.” The angle between the x-axis and the vector is 60 degrees.

Answer

The horizontal and vertical components are

750

ft/sec and

1299.04

ft/sec, respectively.

2-49 §2.1 source ↗

For the following exercises, vector $u$ is given. Find the angle $θ \in [0, 2 π)$ that vector $u$ makes with the positive direction of the x-axis, in a counter-clockwise direction.

[T] Two forces, a horizontal force of $45$ lb and another of $52$ lb, act on the same object. The angle between these forces is $25 ° .$ Find the magnitude and direction angle from the positive x-axis of the resultant force that acts on the object. (Round to two decimal places.)

This figure has two vectors with the same initial point. The first vector is labeled “52 lb” and the second vector is labeled “45 lb.” The angle between the vectors is 25 degrees.

Answer

The magnitude of resultant force is

94.71

lb; the direction angle is

13.42 ° .

2-51 §2.1 source ↗

For the following exercises, vector $u$ is given. Find the angle $θ \in [0, 2 π)$ that vector $u$ makes with the positive direction of the x-axis, in a counter-clockwise direction.

[T] Three forces act on object. Two of the forces have the magnitudes $58$ N and $27$ N, and make angles $53 °$ and $152 °,$ respectively, with the positive x-axis. Find the magnitude and the direction angle from the positive x-axis of the third force such that the resultant force acting on the object is zero. (Round to two decimal places.)

Answer

The magnitude of the third vector is

60.03

N; the direction angle is

259.38 ° .

2-53 §2.1 source ↗

For the following exercises, vector $u$ is given. Find the angle $θ \in [0, 2 π)$ that vector $u$ makes with the positive direction of the x-axis, in a counter-clockwise direction.

[T] An airplane is flying in the direction of $43 °$ east of north (also abbreviated as $N 43 E)$ at a speed of $550$ mph. A wind with speed $25$ mph comes from the southwest at a bearing of $N 15 E .$ What are the ground speed and new direction of the airplane?

This figure is the first quadrant of a coordinate system. There are two vectors both of which have the origin as the initial point. The first vector is labeled “550 miles per hour” and has an angle of 43 degrees from the y-axis. There is also an image of an airplane at the end of the vector. The second vector is labeled “25 miles per hour” and has an angle of 15 degrees from the y-axis.

Answer

The new ground speed of the airplane is

572.19

mph; the new direction is

N 41.82 E .

2-55 §2.1 source ↗

For the following exercises, vector $u$ is given. Find the angle $θ \in [0, 2 π)$ that vector $u$ makes with the positive direction of the x-axis, in a counter-clockwise direction.

[T] A 50-lb weight is hung by a cable so that the two portions of the cable make angles of $40 °$ and $53 °,$ respectively, with the horizontal. Find the magnitudes of the forces of tension $T_{1}$ and $T_{2}$ in the cables if the resultant force acting on the object is zero. (Round to two decimal places.)

This figure is a horizontal line with two vectors below the line formina a triangle with the line. The vectors point toward the ends of the line. The angle between the horizontal line and the first vector is 40 degrees. The angle between the horizontal line and the second vector is 53 degrees. At the point where the two vectors meet is a rectangle labeled “50 lb.”

Answer

‖ T_{1} ‖ =

30.13 lb,

‖ T_{2} ‖ =

38.35 lb

2-57 §2.1 source ↗

For the following exercises, vector $u$ is given. Find the angle $θ \in [0, 2 π)$ that vector $u$ makes with the positive direction of the x-axis, in a counter-clockwise direction.

[T] A 1500-lb boat is parked on a ramp that makes an angle of $30 °$ with the horizontal. The boat’s weight vector points downward and is a sum of two vectors: a horizontal vector $v_{1}$ that is parallel to the ramp and a vertical vector $v_{2}$ that is perpendicular to the inclined surface. The magnitudes of vectors $v_{1}$ and $v_{2}$ are the horizontal and vertical component, respectively, of the boat’s weight vector. Find the magnitudes of $v_{1}$ and $v_{2} .$ (Round to the nearest integer.)

This figure shows a right triangle. The angle between the horizontal base and the hypotenuse is 30 degrees. On the hypotenuse is the image of a boat. From center of the boat there are three vectors. Two of the vectors are orthogonal with one in the direction of the boat and the other below the boat. The third vector is down, perpendicular to the vertical side.

Answer

‖ v_{1} ‖ = 750

lb,

‖ v_{2} ‖ = 1299

2-59 §2.1 source ↗

For the following exercises, vector $u$ is given. Find the angle $θ \in [0, 2 π)$ that vector $u$ makes with the positive direction of the x-axis, in a counter-clockwise direction.

A guy-wire supports a pole that is
$75$ ft high. One end of the wire is attached to the top of the pole and the other end is anchored to the ground $50$ ft from the base of the pole. Determine the horizontal and vertical components of the force of tension in the wire if its magnitude is $50$ lb. (Round to the nearest integer.)

This figure is a tower with a guy wire from the top to the ground. The tower, guy wire, and the ground form a right triangle. The base is labeled “50 feet” and is horizontal. The other side is labeled “75 feet” and is vertical. This side is the tower.

Answer

The two horizontal and vertical components of the force of tension are

28

lb and

42

lb, respectively.

Checkpoint 2.11 §2.2 source ↗

Sketch the point

(−2, 3, −1)

in three-dimensional space.

Answer

This figure is the 3-dimensional coordinate system. In the first octant there is a rectangular solid drawn. One corner is labeled (-2, 3, -1).

Checkpoint 2.12 §2.2 source ↗

Find the distance between points

P_{1} = (1, −5, 4)

and

P_{2} =

(4, −1, −1) .

Answer

5 \sqrt{2}

Checkpoint 2.13 §2.2 source ↗

Write an equation of the plane passing through point

(1, −6, −4)

that is parallel to the xy-plane.

Answer

z = −4

Checkpoint 2.14 §2.2 source ↗

Find the standard equation of the sphere with center

(−2, 4, −5)

containing point

(4, 4, −1) .

Answer

{(x + 2)}^{2} +

{(y - 4)}^{2} +

{(z + 5)}^{2} =

52

Checkpoint 2.15 §2.2 source ↗

Find an equation of the sphere with diameter

P Q,

where

P = (2, −1, −3)

and

Q =

(−2, 5, −1) .

Answer

x^{2} +

{(y - 2)}^{2} +

{(z + 2)}^{2} =

14

Checkpoint 2.16 §2.2 source ↗

Describe the set of points that satisfies

(y + 2) (z - 3) =

0,

and graph the set.

Answer

The set of points forms the two planes $y = −2$ and $z = 3 .$

This figure is the 3-dimensional coordinate system. It has two intersecting planes drawn. The first is the x z-plane. The second is parallel to the y z-plane at the value of z = 3. They are perpendicular to each other.

Checkpoint 2.17 §2.2 source ↗

Describe the set of points in three dimensional space that satisfies

x^{2} +

{(z - 2)}^{2} =

16,

and graph the surface.

Answer

A cylinder of radius 4 centered on the line with $x = 0 and z = 2 .$

This figure is the 3-dimensional coordinate system. It has a cylinder parallel to the y-axis and centered around the y-axis.

Checkpoint 2.18 §2.2 source ↗

Let

S = (3, 8, 2)

and

T = (2, −1, 3) .

Express

\vec{S T}

in component form and in standard unit form.

Answer

\vec{S T} =

〈 −1,

−9,

1 〉 =

− i -

9 j +

k

Checkpoint 2.19 §2.2 source ↗

Let

v = 〈 −1, −1, 1 〉

and

w = 〈 2, 0, 1 〉 .

Find a unit vector in the direction of

5 v + 3 w .

Answer

〈 \frac{1}{3 \sqrt{10}},

-

\frac{5}{3 \sqrt{10}},

\frac{8}{3 \sqrt{10}} 〉

Checkpoint 2.20 §2.2 source ↗

Assume the quarterback and the receiver are in the same place as in the previous example. This time, however, the quarterback throws the ball at velocity of

40

mph and an angle of

45 ° .

Write the initial velocity vector of the ball,

v,

in component form.

Answer

v =

〈 16 \sqrt{2},

12 \sqrt{2},

20 \sqrt{2} 〉

2-61 §2.2 source ↗

Consider a rectangular box with one of the vertices at the origin, as shown in the following figure. If point $A (2, 3, 5)$ is the opposite vertex to the origin, then find

a. the coordinates of the other six vertices of the box and

b. the length of the diagonal of the box determined by the vertices $O$ and $A .$

This figure is the first octant of the 3-dimensional coordinate system. It has a point labeled “A(2, 3, 5)” drawn.

Answer

(2, 0, 5),

(2, 0, 0),

(2, 3, 0),

(0, 3, 0),

(0, 3, 5),

(0, 0, 5);

\sqrt{38}

2-63 §2.2 source ↗

For the following exercises, describe and graph the set of points that satisfies the given equation.

$(y - 5) (z - 6) =$ $0$

Answer

A union of two planes: $y = 5$ (a plane parallel to the xz-plane) and $z = 6$ (a plane parallel to the xy-plane)

This figure is the first octant of the 3-dimensional coordinate system. It has two planes drawn. The first plane is parallel to the x y-plane and is at z = 6. The second plane is parallel to the x z-plane and is at y = 5. The planes are perpendicular.

2-65 §2.2 source ↗

For the following exercises, describe and graph the set of points that satisfies the given equation.

${(y - 1)}^{2} +$ ${(z - 1)}^{2} =$ $1$

Answer

A cylinder of radius $1$ centered on the line $y = 1, z = 1$

This figure is the first octant of the 3-dimensional coordinate system. It has a cylinder drawn. The axis of the cylinder is parallel to the x-axis.

2-67 §2.2 source ↗

For the following exercises, describe and graph the set of points that satisfies the given equation.

Write the equation of the plane passing through point $(1, 1, 1)$ that is parallel to the xy-plane.

Answer

z = 1

2-69 §2.2 source ↗

For the following exercises, describe and graph the set of points that satisfies the given equation.

Find an equation of the plane passing through points $(1, −3, −2),$ $(0, 3, −2),$ and $(1, 0, −2) .$

Answer

z = −2

2-71 §2.2 source ↗

For the following exercises, find an equation of the sphere in standard form that satisfies the given conditions.

Center $C (−1, 7, 4)$ and radius $4$

Answer

{(x + 1)}^{2} +

{(y - 7)}^{2} +

{(z - 4)}^{2} =

16

2-73 §2.2 source ↗

For the following exercises, find an equation of the sphere in standard form that satisfies the given conditions.

Diameter $P Q,$ where $P (−1, 5, 7)$ and $Q (−5, 2, 9)$

Answer

{(x + 3)}^{2} +

{(y - 3.5)}^{2} +

{(z - 8)}^{2} =

\frac{29}{4}

2-75 §2.2 source ↗

For the following exercises, find the center and radius of the sphere with an equation in general form that is given.

$x^{2} +$ $y^{2} +$ $z^{2} -$ $4 z +$ $3 =$ $0$

Answer

Center

C (0, 0, 2)

and radius

1

2-77 §2.2 source ↗

For the following exercises, express vector $\vec{P Q}$ with the initial point at $P$ and the terminal point at $Q$

$P (3, 0, 2)$ and $Q (−1, −1, 4)$

Answer

\vec{P Q} =

〈 −4,

−1,

2 〉;

\vec{P Q} =

−4 i -

j +

2 k

2-79 §2.2 source ↗

For the following exercises, express vector $\vec{P Q}$ with the initial point at $P$ and the terminal point at $Q$

$P (−2, 5, −8)$ and $M (1, −7, 4),$ where $M$ is the midpoint of the line segment $P Q$

Answer

\vec{P Q} =

〈 6,

−24,

24 〉;

\vec{P Q} =

6 i -

24 j +

24 k

2-81 §2.2 source ↗

For the following exercises, express vector $\vec{P Q}$ with the initial point at $P$ and the terminal point at $Q$

Find terminal point $Q$ of vector $\vec{P Q} =$ $〈 7,$ $−1,$ $3 〉$ with the initial point at $P (−2, 3, 5) .$

Answer

Q (5, 2, 8)

2-83 §2.2 source ↗

For the following exercises, use the given vectors $a$ and $b$ to find and express the vectors $a + b,$ $4 a,$ and $−5 a + 3 b$ in component form.

$a =$ $〈 −1,$ $−2,$ $4 〉,$ $b = 〈 −5, 6, −7 〉$

Answer

a +

b =

〈 −6,

4,

−3 〉,

4 a =

〈 −4,

−8,

16 〉,

−5 a +

3 b =

〈 −10,

28,

−41 〉

2-85 §2.2 source ↗

For the following exercises, use the given vectors $a$ and $b$ to find and express the vectors $a + b,$ $4 a,$ and $−5 a + 3 b$ in component form.

$a = − k,$ $b = − i$

Answer

a +

b =

〈 −1,

0,

−1 〉,

4 a =

〈 0,

0,

−4 〉,

−5 a +

3 b =

〈 −3,

0,

5 〉

2-87 §2.2 source ↗

For the following exercises, vectors u and v are given. Find the magnitudes of vectors $u - v$ and $−2 u .$

$u = 2 i + 3 j + 4 k,$ $v = − i + 5 j - k$

Answer

‖ u - v ‖ = \sqrt{38},

‖ −2 u ‖ = 2 \sqrt{29}

2-89 §2.2 source ↗

For the following exercises, vectors u and v are given. Find the magnitudes of vectors $u - v$ and $−2 u .$

$u =$ $〈 2 cos t,$ $−2 sin t,$ $3 〉,$ $v = 〈 0, 0, 3 〉,$ where $t$ is a real number.

Answer

‖ u - v ‖ = 2,

‖ −2 u ‖ = 2 \sqrt{13}

2-91 §2.2 source ↗

For the following exercises, find the unit vector in the direction of the given vector $a$ and express it using standard unit vectors.

$a = 3 i - 4 j$

Answer

a = \frac{3}{5} i - \frac{4}{5} j

2-93 §2.2 source ↗

For the following exercises, find the unit vector in the direction of the given vector $a$ and express it using standard unit vectors.

$a = \vec{P Q},$ where $P (−2, 3, 1)$ and $Q (0, −4, 4)$

Answer

⟨ \frac{2}{\sqrt{62}},

\frac{7}{\sqrt{62}},

\frac{3}{\sqrt{62}} ⟩

2-95 §2.2 source ↗

For the following exercises, find the unit vector in the direction of the given vector $a$ and express it using standard unit vectors.

$a = u - v + w,$ where $u = i - j - k,$ $v = 2 i - j + k,$ and $w = − i + j + 3 k$

Answer

〈 - \frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}} 〉

2-97 §2.2 source ↗

For the following exercises, find the unit vector in the direction of the given vector $a$ and express it using standard unit vectors.

Determine whether $\vec{A B}$ and $\vec{P Q}$ are equivalent vectors, where $A (1, 1, 1),$ $B (3, 3, 3),$ $P (1, 4, 5),$ and $Q (3, 6, 7) .$

Answer

Equivalent vectors

2-99 §2.2 source ↗

For the following exercises, find vector $u$ with a magnitude that is given and satisfies the given conditions.

$v = 〈 7, −1, 3 〉,$ $‖ u ‖ = 10,$ $u$ and $v$ have the same direction

Answer

u =

〈 \frac{70}{\sqrt{59}},

-

\frac{10}{\sqrt{59}},

\frac{30}{\sqrt{59}} 〉

2-101 §2.2 source ↗

For the following exercises, find vector $u$ with a magnitude that is given and satisfies the given conditions.

$v =$ $〈 2 sin t,$ $2 cos t,$ $1 〉,$ $‖ u ‖ = 2,$ $u$ and $v$ have opposite directions for any $t,$ where $t$ is a real number

Answer

u =

〈 -

\frac{4}{\sqrt{5}} sin t,

-

\frac{4}{\sqrt{5}} cos t,

-

\frac{2}{\sqrt{5}} 〉

2-103 §2.2 source ↗

For the following exercises, find vector $u$ with a magnitude that is given and satisfies the given conditions.

Determine a vector of magnitude $5$ in the direction of vector $\vec{A B},$ where $A (2, 1, 5)$ and $B (3, 4, −7) .$

Answer

〈 \frac{5}{\sqrt{154}},

\frac{15}{\sqrt{154}},

-

\frac{60}{\sqrt{154}} 〉

2-105 §2.2 source ↗

For the following exercises, find vector $u$ with a magnitude that is given and satisfies the given conditions.

Consider the points $A (2, α, 0),$ $B (0, 1, β),$ and $C (1, 1, β),$ where $α$ and $β$ are negative real numbers. Find $α$ and $β$ such that $‖ \vec{O A} -$ $\vec{O B} +$ $\vec{O C} ‖ =$ $‖ \vec{O B} ‖ =$ $4 .$

Answer

α = − \sqrt{7},

β = − \sqrt{15}

2-111 §2.2 source ↗

For the following exercises, find vector $u$ with a magnitude that is given and satisfies the given conditions.

[T] A force $F$ of $50 N$ acts on a particle in the direction of the vector $\vec{O P},$ where $P (3, 4, 0) .$

a. Express the force as a vector in component form.

b. Find the angle between force $F$ and the positive direction of the x-axis. Express the answer in degrees rounded to the nearest integer.

Answer

F =

〈 30,

40,

0 〉;

53 °

2-113 §2.2 source ↗

For the following exercises, find vector $u$ with a magnitude that is given and satisfies the given conditions.

If $F$ is a force that moves an object from point $P_{1} (x_{1}, y_{1}, z_{1})$ to another point $P_{2} (x_{2}, y_{2}, z_{2}),$ then the displacement vector is defined as $D =$ $(x_{2} - x_{1}) i +$ $(y_{2} - y_{1}) j +$ $(z_{2} - z_{1}) k .$ A metal container is lifted $10$ m vertically by a constant force $F .$ Express the displacement vector $D$ by using standard unit vectors.

Answer

D = 10 k

2-115 §2.2 source ↗

For the following exercises, find vector $u$ with a magnitude that is given and satisfies the given conditions.

The sum of the forces acting on an object is called the resultant or net force. An object is said to be in static equilibrium if the resultant force of the forces that act on it is zero. Let $F_{1} =$ $〈 10,$ $6,$ $3 〉,$ $F_{2} = 〈 0, 4, 9 〉,$ and $F_{3} =$ $〈 10,$ $−3,$ $−9 〉$ be three forces acting on a box. Find the force $F_{4}$ acting on the box such that the box is in static equilibrium. Express the answer in component form.

Answer

F_{4} =

〈 −20,

−7,

−3 〉

2-117 §2.2 source ↗

For the following exercises, find vector $u$ with a magnitude that is given and satisfies the given conditions.

The force of gravity $F$ acting on an object is given by $F = m g,$ where m is the mass of the object (expressed in kilograms) and $g$ is acceleration resulting from gravity, with $‖ g ‖ = 9.8$ $N/kg .$ A 2-kg disco ball hangs by a chain from the ceiling of a room.

a. Find the force of gravity $F$ acting on the disco ball and find its magnitude.

b. Find the force of tension $T$ in the chain and its magnitude.
Express the answers using standard unit vectors.

This figure shows a disco ball suspended from a ceiling. — Figure 2.43 in the book ↗

Answer

F = −19.6 k,

‖ F ‖ = 19.6

N; b.

T = 19.6 k,

‖ T ‖ = 19.6

2-119 §2.2 source ↗

For the following exercises, find vector $u$ with a magnitude that is given and satisfies the given conditions.

[T] A 30-kg block of cement is suspended by three cables of equal length that are anchored at points $P (−2, 0, 0),$ $Q (1, \sqrt{3}, 0),$ and $R (1, − \sqrt{3}, 0) .$ The load is located at $S (0, 0, −2 \sqrt{3}),$ as shown in the following figure. Let $F_{1},$ $F_{2},$ and $F_{3}$ be the forces of tension resulting from the load in cables $R S, Q S,$ and $P S,$ respectively.

a. Find the gravitational force $F$ acting on the block of cement that counterbalances the sum $F_{1} + F_{2} + F_{3}$ of the forces of tension in the cables.

b. Find forces $F_{1},$ $F_{2},$ and $F_{3} .$ Express the answer in component form.

This figure is the 3-dimensional coordinate system. It has 4 points drawn. The first point is labeled “P(-2, 0, 0).” The second point is labeled “R(1, -squareroot of 3, 0).” The third point is labeled “S(0, 0, -2squareroots of 3).” The fourth point is labeled “Q(1, squareroot of 3, 0).” There are line segments from P to S, from R to S, and from Q to S. At point S there is a box labeled “30 k g.”

Answer

F = −294 k

N; b.

F_{1} =

〈 -

\frac{49 \sqrt{3}}{3},

49,

−98 〉,

F_{2} =

〈 -

\frac{49 \sqrt{3}}{3},

−49,

−98 〉,

and

F_{3} =

〈 \frac{98 \sqrt{3}}{3},

0,

−98 〉

(each component is expressed in newtons)

2-121 §2.2 source ↗

For the following exercises, find vector $u$ with a magnitude that is given and satisfies the given conditions.

Let $r (t) =$ $〈 x (t),$ $y (t),$ $z (t) 〉$ be the position vector of a particle at the time $t \in [0, T],$ where $x, y,$ and $z$ are smooth functions on $[0, T] .$ The instantaneous velocity of the particle at time $t$ is defined by vector $v (t) =$ $〈 x ′ (t),$ $y ′ (t),$ $z ′ (t) 〉,$ with components that are the derivatives with respect to $t,$ of the functions x, y, and z, respectively. The magnitude $‖ v (t) ‖$ of the instantaneous velocity vector is called the speed of the particle at time t. Vector $a (t) =$ $〈 x^{″} (t),$ $y^{″} (t),$ $z^{″} (t) 〉,$ with components that are the second derivatives with respect to $t,$ of the functions $x, y,$ and $z,$ respectively, gives the acceleration of the particle at time $t .$ Consider $r (t) =$ $〈 cos t,$ $sin t,$ $2 t 〉$ the position vector of a particle at time $t \in [0, 30],$ where the components of $r$ are expressed in centimeters and time is expressed in seconds.

a. Find the instantaneous velocity, speed, and acceleration of the particle after the first second. Round your answer to two decimal places.

b. Use a CAS to visualize the path of the particle—that is, the set of all points of coordinates $(cos t, sin t, 2 t),$ where $t \in [0, 30] .$

Answer

a. $v (1) =$ $〈 −0.84,$ $0.54,$ $2 〉$ (each component is expressed in centimeters per second); $‖ v (1) ‖ = 2.24$ (expressed in centimeters per second); $a (1) =$ $〈 −0.54,$ $−0.84,$ $0 〉$ (each component expressed in centimeters per second squared);

This figure is of the 3-dimensional coordinate system above the xy-plane. It has a spiral drawn resembling a spring. The spiral is around the z-axis. The spiral starts on the x-axis at x = 1.

Checkpoint 2.21 §2.3 source ↗

Find

u \cdot v,

where

u = 〈 2, 9, −1 〉

and

v =

〈 −3,

1,

−4 〉 .

Answer

Checkpoint 2.22 §2.3 source ↗

Find the following products for $p = 〈 7, 0, 2 〉,$ $q =$ $〈 −2,$ $2,$ $−2 〉,$ and $r = 〈 0, 2, −3 〉 .$

a. $(r \cdot p) q$

b. ${‖ p ‖}^{2}$

Answer

(r \cdot p) q =

〈 12,

−12,

12 〉;

{‖ p ‖}^{2} = 53

Checkpoint 2.23 §2.3 source ↗

Find the measure of the angle, in radians, formed by vectors

a = 〈 1, 2, 0 〉

and

b = 〈 2, 4, 1 〉 .

Round to the nearest hundredth.

Answer

θ \approx 0.22

rad

Checkpoint 2.24 §2.3 source ↗

For which value of x is

p = 〈 2, 8, −1 〉

orthogonal to

q = 〈 x, −1, 2 〉 ?

Answer

x = 5

Checkpoint 2.25 §2.3 source ↗

Let $v = 〈 3, −5, 1 〉 .$ Find the measure of the angles formed by each pair of vectors.

a. $v$ and i

b. $v$ and j

c. $v$ and k

Answer

α \approx 1.04

rad; b.

β \approx 2.58

rad; c.

γ \approx 1.40

rad

Checkpoint 2.26 §2.3 source ↗

On June 1, AAA Party Supply Store decided to increase the price they charge for party favors to $2 per package. They also changed suppliers for their invitations, and are now able to purchase invitations for only 10¢ per package. All their other costs and prices remain the same. If AAA sells 1408 invitations, 147 party favors, 2112 decorations, and 1894 food service items in the month of June, use vectors and dot products to calculate their total sales and profit for June.

Answer

Sales = $15,685.50; profit = $14,073.15

Checkpoint 2.27 §2.3 source ↗

Express

v = 5 i - j

as a sum of orthogonal vectors such that one of the vectors has the same direction as

u = 4 i + 2 j .

Answer

v = p + q,

where

p = \frac{18}{5} i + \frac{9}{5} j

and

q = \frac{7}{5} i - \frac{14}{5} j

Checkpoint 2.28 §2.3 source ↗

Repeat the previous example, but assume the ocean current is moving southeast instead of northeast, as shown in the following figure.

This figure is an image of a ship. The ship is at the origin of two perpendicular axes. The horizontal axis is labeled “east.” The second axis is vertical and labeled “north.” From the ship there are two vectors. The first is labeled “v” and has an angle of 15 degrees between the East axis and the vector v. The second vector is labeled “w” and has an angle of 45 degrees between the East axis and the vector w. Vector w is below the East axis in the fourth quadrant.

Answer

21 knots

Checkpoint 2.29 §2.3 source ↗

A constant force of 30 lb is applied at an angle of 60° to pull a handcart 10 ft across the ground (Figure 2.52). What is the work done by this force?

This figure is an image of a hand cart with a crate. The vertical handle of the hand cart has two vectors. The first is horizontal to the handle and labeled “s.” The second is from the handle and labeled “F.” The angle between the two vectors is 60 degrees. — Figure 2.52 in the book ↗

Answer

150 ft-lb

2-123 §2.3 source ↗

For the following exercises, the vectors $u$ and $v$ are given. Calculate the dot product $u \cdot v .$

$u = 〈 3, 0 〉,$ $v = 〈 2, 2 〉$

Answer

2-125 §2.3 source ↗

For the following exercises, the vectors $u$ and $v$ are given. Calculate the dot product $u \cdot v .$

$u = 〈 2, 2, −1 〉,$ $v = 〈 −1, 2, 2 〉$

Answer

2-127 §2.3 source ↗

For the following exercises, the vectors a, b, and c are given. Determine the vectors $(a \cdot b) c$ and $(a \cdot c) b .$ Express the vectors in component form.

$a = 〈 2, 0, −3 〉,$ $b =$ $〈 −4,$ $−7,$ $1 〉,$ $c = 〈 1, 1, −1 〉$

Answer

(a \cdot b) c =

〈 −11,

−11,

11 〉;

(a \cdot c) b =

〈 −20,

−35,

5 〉

2-129 §2.3 source ↗

For the following exercises, the vectors a, b, and c are given. Determine the vectors $(a \cdot b) c$ and $(a \cdot c) b .$ Express the vectors in component form.

$a = i + j,$ $b = i - k,$ $c = i - 2 k$

Answer

(a \cdot b) c =

〈 1,

0,

−2 〉;

(a \cdot c) b =

〈 1,

0,

−1 〉

2-131 §2.3 source ↗

For the following exercises, the two-dimensional vectors a and b are given.

[T] $a = 〈 3, −1 〉,$ $b = 〈 −4, 0 〉$

Answer

θ = 2.82

rad; b.

θ

is not acute.

2-133 §2.3 source ↗

For the following exercises, the two-dimensional vectors a and b are given.

$u = 3 i,$ $v = 4 i + 4 j$

Answer

θ = \frac{π}{4}

rad; b.

θ

is acute.

2-135 §2.3 source ↗

For the following exercises, find the measure of the angle between the three-dimensional vectors a and b. Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly.

$a = 〈 3, −1, 2 〉,$ $b = 〈 1, −1, −2 〉$

Answer

θ = \frac{π}{2}

2-137 §2.3 source ↗

$a = i + j,$ $b = j - k$

Answer

θ = \frac{π}{3}

2-139 §2.3 source ↗

[T] $a = 3 i - j - 2 k,$ $b = v + w,$ where $v = −2 i - 3 j + 2 k$ and $w = i + 2 k$

Answer

θ = 2

rad

2-141 §2.3 source ↗

For the following exercises determine whether the given vectors are orthogonal.

$a = 〈 x, y 〉,$ $b = 〈 − y, x 〉,$ where x and y are nonzero real numbers

Answer

Orthogonal

2-143 §2.3 source ↗

For the following exercises determine whether the given vectors are orthogonal.

$a = 3 i - j - 2 k,$ $b = −2 i - 3 j + k$

Answer

Not orthogonal

2-145 §2.3 source ↗

For the following exercises determine whether the given vectors are orthogonal.

Find all two-dimensional vectors a orthogonal to vector $b = 〈 3, 4 〉 .$ Express the answer in component form.

Answer

a = 〈 - \frac{4 α}{3}, α 〉,

where

α \neq 0

is a real number

2-147 §2.3 source ↗

For the following exercises determine whether the given vectors are orthogonal.

Determine all three-dimensional vectors $u$ orthogonal to vector $v = 〈 1, 1, 0 〉 .$ Express the answer by using standard unit vectors.

Answer

u =

− α i +

α j +

β k,

where

α

and

β

are real numbers such that

α^{2} + β^{2} \neq 0

2-149 §2.3 source ↗

For the following exercises determine whether the given vectors are orthogonal.

Determine the real number $α$ such that vectors $a = 2 i + 3 j$ and $b = 9 i + α j$ are orthogonal.

Answer

α = −6

2-151 §2.3 source ↗

For the following exercises determine whether the given vectors are orthogonal.

[T] Consider the points $P (4, 5)$ and $Q (5, −7) .$

a. Determine vectors $\vec{O P}$ and $\vec{O Q} .$ Express the answer by using standard unit vectors.

b. Determine the measure of angle O in triangle OPQ. Express the answer in degrees rounded to two decimal places.

Answer

\vec{O P} = 4 i + 5 j,

\vec{O Q} = 5 i - 7 j;

105.8 °

2-153 §2.3 source ↗

For the following exercises determine whether the given vectors are orthogonal.

Determine the measure of angle A in triangle ABC, where $A (1, 1, 8),$ $B (4, −3, −4),$ and $C (−3, 1, 5) .$ Express your answer in degrees rounded to two decimal places.

Answer

68.33 °

2-155 §2.3 source ↗

For the following exercises, determine which (if any) pairs of the following vectors are orthogonal.

$u = 〈 3, 7, −2 〉,$ $v =$ $〈 5,$ $−3,$ $−3 〉,$ $w = 〈 0, 1, −1 〉$

Answer

u

and

v

are orthogonal;

v

and

w

are orthogonal.

2-161 §2.3 source ↗

For the following problems, the vector $u$ is given.

$u = 〈 2, 2, 1 〉$

Answer

cos α =

\frac{2}{3},

cos β =

\frac{2}{3},

and

cos γ = \frac{1}{3};

α = 48 °,

β = 48 °,

and

γ = 71 °

2-163 §2.3 source ↗

For the following problems, the vector $u$ is given.

$u = 〈 −1, 5, 2 〉$

Answer

cos α =

-

\frac{1}{\sqrt{30}},

cos β =

\frac{5}{\sqrt{30}},

and

cos γ = \frac{2}{\sqrt{30}};

α = 101 °,

β = 24 °,

and

γ = 69 °

2-167 §2.3 source ↗

For the following exercises, the vectors $u$ and $v$ are given.

$u = 5 i + 2 j,$ $v = 2 i + 3 j$

Answer

w =

〈 \frac{80}{29},

\frac{32}{29} 〉;

{comp}_{u} v = \frac{16}{\sqrt{29}}

2-169 §2.3 source ↗

For the following exercises, the vectors $u$ and $v$ are given.

$u = 3 i + 2 k,$ $v = 2 j + 4 k$

Answer

w =

〈 \frac{24}{13},

0,

\frac{16}{13} 〉;

{comp}_{u} v = \frac{8}{\sqrt{13}}

2-171 §2.3 source ↗

For the following exercises, the vectors $u$ and $v$ are given.

Consider the vectors $u = 4 i - 3 j$ and $v = 3 i + 2 j .$

a. Find the component form of vector $w = {proj}_{u} v$ that represents the projection of $v$ onto $u$ .

b. Write the decomposition $v = w + q$ of vector $v$ into the orthogonal components $w$ and $q$ , where $w$ is the projection of $v$ onto $u$ and $q$ is a vector orthogonal to the direction of $u$ .

Answer

w =

〈 \frac{24}{25},

-

\frac{18}{25} 〉;

q =

〈 \frac{51}{25},

\frac{68}{25} 〉,

v =

w +

q =

〈 \frac{24}{25},

-

\frac{18}{25} 〉 +

〈 \frac{51}{25},

\frac{68}{25} 〉

2-173 §2.3 source ↗

For the following exercises, the vectors $u$ and $v$ are given.

A methane molecule has a carbon atom situated at the origin and four hydrogen atoms located at points $P (1, 1, −1),$ $Q (1, −1, 1),$ $R (−1, 1, 1),$ $and S (−1, −1, −1)$ (see figure).

a. Find the distance between the hydrogen atoms located at P and R.

b. Find the angle between vectors $\vec{O S}$ and $\vec{O R}$ that connect the carbon atom with the hydrogen atoms located at S and R, which is also called the bond angle. Express the answer in degrees rounded to two decimal places.

This figure is the 3-dimensional coordinate system. There are four points plotted. The first point is labeled “P(1, 1, -1),” the second point is labeled “Q(1, -1, 1),” the third point is labeled “R(-1, 1, 1),” and the fourth point is labeled “S(-1, -1, -1).” There are line segments from Q to P, P to R and R to P. There are also two vectors in standard position. The first has terminal point of R and the second has terminal point of S. The angle between them is represented with an arc.

Answer

2 \sqrt{2};

109.47 °

2-175 §2.3 source ↗

For the following exercises, the vectors $u$ and $v$ are given.

Find the work done by force $F = 〈 5, 6, −2 〉$ (measured in Newtons) that moves a particle from point $P (3, −1, 0)$ to point $Q (2, 3, 1)$ along a straight line (the distance is measured in meters).

Answer

17 N \cdot m

2-177 §2.3 source ↗

For the following exercises, the vectors $u$ and $v$ are given.

[T] A father is pulling his son on a sled at an angle of $20 °$ with the horizontal with a force of 25 lb (see the following image). He pulls the sled in a straight path of 50 ft. How much work was done by the man pulling the sled? (Round the answer to the nearest integer.)

This figure is an image of a person pulling a child on a sled. The rope for pulling the sled is represented by a vector and labeled “25 lb.” There is an angle between the rope vector and the horizontal ground of 20 degrees.

Answer

1175

ft \cdot lb

2-179 §2.3 source ↗

For the following exercises, the vectors $u$ and $v$ are given.

[T] A boat sails north aided by a wind blowing in a direction of $N3 0 ° E$ with a magnitude of 500 lb. How much work is performed by the wind as the boat moves 100 ft? (Round the answer to two decimal places.)

Answer

W = 43301.27

ft-lb

2-181 §2.3 source ↗

For the following exercises, the vectors $u$ and $v$ are given.

[T] Two forces $F_{1}$ and $F_{2}$ are represented by vectors with initial points that are at the origin. The first force has a magnitude of 20 lb and the terminal point of the vector is point $P (1, 1, 0) .$ The second force has a magnitude of 40 lb and the terminal point of its vector is point $Q (0, 1, 1) .$ Let F be the resultant force of forces $F_{1}$ and $F_{2} .$

a. Find the magnitude of F. (Round the answer to one decimal place.)

b. Find the direction angles of F. (Express the answer in degrees rounded to one decimal place.)

Answer

‖ F_{1} +

F_{2} ‖ =

52.9

lb; b. The direction angles are

α = 74.5 °,

β = 36.7 °,

and

γ = 57.7 ° .

Checkpoint 2.30 §2.4 source ↗

Find

p \times q

for

p = 〈 5, 1, 2 〉

and

q = 〈 −2, 0, 1 〉 .

Express the answer using standard unit vectors.

Answer

i - 9 j + 2 k

Checkpoint 2.31 §2.4 source ↗

Suppose vectors

u

and

v

lie in the xy-plane (the z-component of each vector is zero). Now suppose the x- and y-components of

u

and the y-component of

v

are all positive, whereas the x-component of

v

is negative. Assuming the coordinate axes are oriented in the usual positions, in which direction does

u \times v

point?

Answer

Up (the positive z-direction)

Checkpoint 2.32 §2.4 source ↗

Find

(i \times j) \times (k \times i) .

Answer

− i

Checkpoint 2.33 §2.4 source ↗

Use the properties of the cross product to calculate

(i \times k) \times (k \times j) .

Answer

− k

Checkpoint 2.34 §2.4 source ↗

Use Properties of the Cross Product to find the magnitude of

u \times v,

where

u = 〈 −8, 0, 0 〉

and

v = 〈 0, 2, 0 〉 .

Answer

16

Checkpoint 2.35 §2.4 source ↗

Evaluate the determinant

| \begin{matrix} 1 & −2 & −1 \\ 3 & 2 & −3 \\ 1 & 5 & 4 \end{matrix} | .

Answer

40

Checkpoint 2.36 §2.4 source ↗

Use determinant notation to find

a \times b,

where

a = 〈 8, 2, 3 〉

and

b = 〈 −1, 0, 4 〉 .

Answer

8 i - 35 j + 2 k

Checkpoint 2.37 §2.4 source ↗

Find a unit vector orthogonal to both

a

and

b,

where

a = 〈 4, 0, 3 〉

and

b = 〈 1, 1, 4 〉 .

Answer

〈 \frac{−3}{\sqrt{194}},

\frac{−13}{\sqrt{194}},

\frac{4}{\sqrt{194}} 〉

Checkpoint 2.38 §2.4 source ↗

Find the area of the parallelogram

P Q R S

with vertices

P (1, 1, 0),

Q (7, 1, 0),

R (9, 4, 2),

and

S (3, 4, 2) .

Answer

6 \sqrt{13}

Checkpoint 2.39 §2.4 source ↗

Calculate the triple scalar product

a \cdot (b \times c),

where

a = 〈 2, −4, 1 〉,

b = 〈 0, 3, −1 〉,

and

c = 〈 5, −3, 3 〉 .

Answer

17

Checkpoint 2.40 §2.4 source ↗

Find the volume of the parallelepiped formed by the vectors

a = 3 i + 4 j - k,

b = 2 i - j - k,

and

c = 3 j + k .

Answer

8

units³

Checkpoint 2.41 §2.4 source ↗

Are the vectors

a = i + j - k,

b = i - j + k,

and

c = i + j + k

coplanar?

Answer

No, the triple scalar product is

−4 \neq 0,

so the three vectors form the adjacent edges of a parallelepiped. They are not coplanar.

Checkpoint 2.42 §2.4 source ↗

Calculate the force required to produce

15 N \cdot m

torque at an angle of

30 º

from a 150-cm rod.

Answer

20

2-183 §2.4 source ↗

For the following exercises, the vectors $u$ and $v$ are given.

$u = 〈 2, 0, 0 〉,$ $v = 〈 2, 2, 0 〉$

Answer

a. $u \times v =$ $〈 0,$ $0,$ $4 〉;$
b.

This figure is the first octant of the 3-dimensional coordinate system. On the x-axis there is a vector labeled “u.” In the x y-plane there is a vector labeled “v.” On the z-axis there is the vector labeled “u cross v.”

2-185 §2.4 source ↗

For the following exercises, the vectors $u$ and $v$ are given.

$u = 2 i + 3 j,$ $v = j + 2 k$

Answer

a. $u \times v =$ $〈 6,$ $−4,$ $2 〉;$
b.

This figure is the first octant of the 3-dimensional coordinate system and shows three vectors. The first vector is labeled u and has components less than 2, 3, 0 greater than . The second vector is labeled v and has components less than 0, 1, 2 greater than .” The third vector is labeled u cross v and has components less than 6, -4, 2 greater than .”

2-187 §2.4 source ↗

For the following exercises, the vectors $u$ and $v$ are given.

Simplify $(i \times i - 2 i \times j - 4 i \times k + 3 j \times k) \times i .$

Answer

−2 j - 4 k

2-189 §2.4 source ↗

In the following exercises, vectors $u$ and $v$ are given. Find unit vector $w$ in the direction of the cross product vector $u \times v .$ Express your answer using standard unit vectors.

$u = 〈 3, −1, 2 〉,$ $v = 〈 −2, 0, 1 〉$

Answer

w =

-

\frac{1}{3 \sqrt{6}} i -

\frac{7}{3 \sqrt{6}} j -

\frac{2}{3 \sqrt{6}} k

2-191 §2.4 source ↗

In the following exercises, vectors $u$ and $v$ are given. Find unit vector $w$ in the direction of the cross product vector $u \times v .$ Express your answer using standard unit vectors.

$u = \vec{A B},$ $v = \vec{A C},$ where $A (1, 0, 1),$ $B (1, −1, 3),$ and $C (0, 0, 5)$

Answer

w =

-

\frac{4}{\sqrt{21}} i -

\frac{2}{\sqrt{21}} j -

\frac{1}{\sqrt{21}} k

2-193 §2.4 source ↗

In the following exercises, vectors $u$ and $v$ are given. Find unit vector $w$ in the direction of the cross product vector $u \times v .$ Express your answer using standard unit vectors.

Determine the real number $α$ such that $u \times v$ and $i$ are orthogonal, where $u = 3 i + j - 5 k$ and $v = 4 i - 2 j + α k .$

Answer

α = 10

2-197 §2.4 source ↗

In the following exercises, vectors $u$ and $v$ are given. Find unit vector $w$ in the direction of the cross product vector $u \times v .$ Express your answer using standard unit vectors.

Calculate the determinant $| \begin{matrix} i & j & k \\ 1 & −1 & 7 \\ 2 & 0 & 3 \end{matrix} | .$

Answer

−3 i + 11 j + 2 k

2-199 §2.4 source ↗

For the following exercises, the vectors $u$ and $v$ are given. Use determinant notation to find vector $w$ orthogonal to vectors $u$ and $v .$

$u =$ $〈 −1,$ $0,$ $e^{t} 〉,$ $v =$ $〈 1,$ $e^{− t},$ $0 〉,$ where $t$ is a real number

Answer

w =

〈 −1,

e^{t},

− e^{− t} 〉

2-201 §2.4 source ↗

For the following exercises, the vectors $u$ and $v$ are given. Use determinant notation to find vector $w$ orthogonal to vectors $u$ and $v .$

Find vector $(a - 2 b) \times c,$ where $a = | \begin{matrix} i & j & k \\ 2 & −1 & 5 \\ 0 & 1 & 8 \end{matrix} |,$ $b = | \begin{matrix} i & j & k \\ 0 & 1 & 1 \\ 2 & −1 & −2 \end{matrix} |,$ and $c = i + j + k .$

Answer

−26 i + 17 j + 9 k

2-203 §2.4 source ↗

For the following exercises, the vectors $u$ and $v$ are given. Use determinant notation to find vector $w$ orthogonal to vectors $u$ and $v .$

[T] Use the cross product $u \times v$ to find the acute angle between vectors $u$ and $v,$ where $u = i + 2 j$ and $v = i + k .$ Express the answer in degrees rounded to the nearest integer.

Answer

72 °

2-209 §2.4 source ↗

For the following exercises, the vectors $u$ and $v$ are given. Use determinant notation to find vector $w$ orthogonal to vectors $u$ and $v .$

Find the area of the parallelogram with adjacent sides $u = 〈 3, 2, 0 〉$ and $v = 〈 0, 2, 1 〉 .$

Answer

7

2-211 §2.4 source ↗

For the following exercises, the vectors $u$ and $v$ are given. Use determinant notation to find vector $w$ orthogonal to vectors $u$ and $v .$

Consider points $A (3, −1, 2),$ $B (2, 1, 5),$ and $C (1, −2, −2) .$

a. Find the area of parallelogram $A B C D$ with adjacent sides $\vec{A B}$ and $\vec{A C} .$

b. Find the area of triangle $A B C .$

c. Find the distance from point $A$ to line $B C .$

Answer

5 \sqrt{6};

\frac{5 \sqrt{6}}{2};

\frac{5 \sqrt{6}}{\sqrt{59}}

2-213 §2.4 source ↗

In the following exercises, vectors $u, v, and w$ are given.

$u = i + j,$ $v = j + k,$ and $w = i + k$

Answer

2;

2

2-215 §2.4 source ↗

In the following exercises, vectors $u, v, and w$ are given.

Calculate the triple scalar products $v \cdot (u \times w)$ and $w \cdot (u \times v),$ where $u = 〈 1, 1, 1 〉,$ $v = 〈 7, 6, 9 〉,$ and $w = 〈 4, 2, 7 〉 .$

Answer

v \cdot (u \times w) = −1,

w \cdot (u \times v) = 1

2-217 §2.4 source ↗

In the following exercises, vectors $u, v, and w$ are given.

Find vectors $a, b, and c$ with a triple scalar product given by the determinant

$| \begin{matrix} 1 & 2 & 3 \\ 0 & 2 & 5 \\ 8 & 9 & 2 \end{matrix} | .$ Determine their triple scalar product.

Answer

a = 〈 1, 2, 3 〉,

b = 〈 0, 2, 5 〉,

c = 〈 8, 9, 2 〉;

a \cdot (b \times c) = −9

2-219 §2.4 source ↗

In the following exercises, vectors $u, v, and w$ are given.

Consider the parallelepiped with edges $O A, O B,$ and $O C,$ where $A (2, 1, 0),$ $B (1, 2, 0),$ and $C (0, 1, α) .$

a. Find the real number $α > 0$ such that the volume of the parallelepiped is $3$ units³.

b. For $α = 1,$ find the height $h$ from vertex $C$ of the parallelepiped to the plane formed by the edges $O A$ and $O B$ .

Answer

a. $α = 1;$ b. $h = 1,$

This figure is the first octant of the 3-dimensional coordinate system. There is a parallelepided drawn. From the origin there are three vectors to vertices on the parallelepiped. They are vectors to the points A (2, 1, 0); B (1, 2, 0); and C (0, 1, alpha).

2-225 §2.4 source ↗

In the following exercises, vectors $u, v, and w$ are given.

Consider points $A (0, 0, 2),$ $B (1, 0, 2),$ $C (1, 1, 2),$ and $D (0, 1, 2) .$ Are vectors $\vec{A B},$ $\vec{A C},$ and $\vec{A D}$ linearly dependent (that is, one of the vectors is a linear combination of the other two)?

Answer

Yes,

\vec{A D} =

α \vec{A B} +

β \vec{A C},

where

α = −1

and

β = 1 .

2-227 §2.4 source ↗

In the following exercises, vectors $u, v, and w$ are given.

Let $u = 〈 u_{1}, u_{2} 〉$ and $v = 〈 v_{1}, v_{2} 〉$ be two-dimensional vectors. The cross product of vectors $u$ and $v$ is not defined. However, if the vectors are regarded as the three-dimensional vectors $\tilde{u} =$ $〈 u_{1},$ $u_{2},$ $0 〉$ and $\tilde{v} =$ $〈 v_{1},$ $v_{2},$ $0 〉,$ respectively, then, in this case, we can define the cross product of $\tilde{u}$ and $\tilde{v} .$ In particular, in determinant notation, the cross product of $\tilde{u}$ and $\tilde{v}$ is given by

\tilde{u} \times \tilde{v} = | \begin{matrix} i & j & k \\ u_{1} & u_{2} & 0 \\ v_{1} & v_{2} & 0 \end{matrix} | .

Use this result to compute $(i cos θ + j sin θ) \times (i s i n θ - j c o s θ),$ where $θ$ is a real number.

Answer

− k

2-229 §2.4 source ↗

In the following exercises, vectors $u, v, and w$ are given.

Determine a vector of magnitude $10$ perpendicular to the plane passing through the x-axis and point $P (1, 2, 4) .$

Answer

〈 0, ± 4 \sqrt{5}, \mp 2 \sqrt{5} 〉

2-233 §2.4 source ↗

In the following exercises, vectors $u, v, and w$ are given.

Find all vectors $w =$ $〈 w_{1},$ $w_{2},$ $w_{3} 〉$ that satisfy the equation $〈 1,$ $1,$ $1 〉 \times w =$ $〈 −1,$ $−1,$ $2 〉 .$

Answer

w =

〈 w_{3} -

1,

w_{3} +

1,

w_{3} 〉,

where

w_{3}

is any real number

2-235 §2.4 source ↗

In the following exercises, vectors $u, v, and w$ are given.

[T] A mechanic uses a 12-in. wrench to turn a bolt. The wrench makes a $30 °$ angle with the horizontal. If the mechanic applies a vertical force of $10$ lb on the wrench handle, what is the magnitude of the torque at point $P$ (see the following figure)? Express the answer in foot-pounds rounded to two decimal places.

This figure is the image of an open-end wrench. The lower portion of the wrench is at point P. The wrench has a length of “12 I n.” The angle the wrench makes with a horizontal line from P is 30 degrees. At the top of the wrench is a downward vertical vector labeled “10 l b.”

Answer

8.66 ft-lb

2-237 §2.4 source ↗

In the following exercises, vectors $u, v, and w$ are given.

[T] Find the magnitude of the force that needs to be applied to the end of a 20-cm wrench located on the positive direction of the y-axis if the force is applied in the direction $〈 0, 1, −2 〉$ and it produces a $100$ N·m torque to the bolt located at the origin.

Answer

559 N

2-239 §2.4 source ↗

In the following exercises, vectors $u, v, and w$ are given.

[T] The force vector $F$ acting on a proton with an electric charge of $1.6 \times 10^{−19} C$ (in coulombs) moving in a magnetic field $B$ where the velocity vector $v$ is given by $F =$ $1.6 \times 10^{−19} (v \times B)$ (here, $v$ is expressed in meters per second, $B$ is in tesla [T], and $F$ is in newtons [N]). Find the force that acts on a proton that moves in the xy-plane at velocity $v = 10^{5} i + 10^{5} j$ (in meters per second) in a magnetic field given by $B = 0.3 j .$

Answer

F =

4.8 \times 10^{−15} k N

2-241 §2.4 source ↗

In the following exercises, vectors $u, v, and w$ are given.

[T] Consider $r (t) =$ $〈 cos t,$ $sin t,$ $2 t 〉$ the position vector of a particle at time $t \in [0, 30],$ where the components of $r$ are expressed in centimeters and time in seconds. Let $\vec{O P}$ be the position vector of the particle after $1$ sec.

a. Determine unit vector $B (t)$ (called the binormal unit vector) that has the direction of cross product vector $v (t) \times a (t),$ where $v (t)$ and $a (t)$ are the instantaneous velocity vector and, respectively, the acceleration vector of the particle after $t$ seconds.

b. Use a CAS to visualize vectors $v (1),$ $a (1),$ and $B (1)$ as vectors starting at point $P$ along with the path of the particle.

Answer

a. $B (t) =$ $〈 \frac{2 sin t}{\sqrt{5}},$ $-$ $\frac{2 cos t}{\sqrt{5}},$ $\frac{1}{\sqrt{5}} 〉;$
b.

This figure is the first octant of the 3-dimensional coordinate system. There is a curve sketched that is increasing. On the curve is a point labeled “P.” At P there is a tangent vector to the curve labeled “v(1).” Also from P there is a vector towards the inside of the curve labeled “a(1).” Finally, there is a vector from P labeled “B(1)” pointing towards the z-axis.

Checkpoint 2.43 §2.5 source ↗

Find parametric and symmetric equations of the line passing through points

(1, −3, 2)

and

(5, −2, 8) .

Answer

Possible set of parametric equations: $x =$ $1 +$ $4 t,$ $y =$ $−3 +$ $t,$ $z =$ $2 +$ $6 t;$

related set of symmetric equations: $\frac{x - 1}{4} =$ $y +$ $3 =$ $\frac{z - 2}{6}$

Checkpoint 2.44 §2.5 source ↗

Find parametric equations of the line segment between points

P (−1, 3, 6)

and

Q (−8, 2, 4) .

Answer

x =

−1 -

7 t,

y =

3 -

t,

z =

6 -

2 t,

0 \leq t \leq 1

Checkpoint 2.45 §2.5 source ↗

Find the distance between point

(0, 3, 6)

and the line with parametric equations

x =

1 -

t,

y =

1 +

2 t,

z =

5 +

3 t .

Answer

\sqrt{\frac{10}{7}}

Checkpoint 2.46 §2.5 source ↗

Describe the relationship between the lines with the following parametric equations:

x = 1 - 4 t, y = 3 + t, z = 8 - 6 t

x = 2 + 3 s, y = 2 s, z = −1 - 3 s .

Answer

These lines are skew because their direction vectors are not parallel and there is no point

(x, y, z)

that lies on both lines.

Checkpoint 2.47 §2.5 source ↗

Find an equation of the plane containing the lines $L_{1}$ and $L_{2} :$

\begin{matrix} L_{1} : x = − y = z \\ L_{2} : \frac{x - 3}{2} = y = z - 2. \end{matrix}

Answer

−2 (x - 1) +

(y + 1) +

3 (z - 1) =

0

−2 x + y + 3 z = 0

Checkpoint 2.48 §2.5 source ↗

Find the distance between point

P = (5, −1, 0)

and the plane given by

4 x + 2 y - z = 3 .

Answer

\frac{15}{\sqrt{21}}

Checkpoint 2.49 §2.5 source ↗

Find parametric equations for the line formed by the intersection of planes

x + y - z = 3

and

3 x - y + 3 z = 5 .

Answer

x =

t,

y =

7 -

3 t,

z =

4 -

2 t

Checkpoint 2.50 §2.5 source ↗

Find the measure of the angle between planes

x + y - z = 3

and

3 x - y + 3 z = 5 .

Give the answer in radians and round to two decimal places.

Answer

1.44

rad

Checkpoint 2.51 §2.5 source ↗

Find the distance between parallel planes

5 x - 2 y + z = 6

and

5 x - 2 y + z = −3 .

Answer

\frac{9}{\sqrt{30}}

2-243 §2.5 source ↗

In the following exercises, points $P$ and $Q$ are given. Let $L$ be the line passing through points $P$ and $Q .$

$P (−3, 5, 9),$ $Q (4, −7, 2)$

Answer

r =

〈 −3,

5,

9 〉 +

t 〈 7,

−12,

−7 〉,

t \in ℝ;

x =

−3 +

7 t,

y =

5 -

12 t,

z =

9 -

7 t,

t \in ℝ;

\frac{x + 3}{7} =

\frac{y - 5}{−12} =

\frac{z - 9}{−7};

x =

−3 +

7 t,

y =

5 -

12 t,

z =

9 -

7 t,

t \in [0, 1]

2-245 §2.5 source ↗

In the following exercises, points $P$ and $Q$ are given. Let $L$ be the line passing through points $P$ and $Q .$

$P (−1, 0, 5),$ $Q (4, 0, 3)$

Answer

r =

〈 −1,

0,

5 〉 +

t 〈 5,

0,

−2 〉,

t \in ℝ;

x =

−1 +

5 t,

y =

0,

z =

5 -

2 t,

t \in ℝ;

\frac{x + 1}{5} =

\frac{z - 5}{−2},

y =

0;

x =

−1 +

5 t,

y =

0,

z =

5 -

2 t,

t \in [0, 1]

2-247 §2.5 source ↗

For the following exercises, point $P$ and vector $v$ are given. Let $L$ be the line passing through point $P$ with direction $v .$

$P (1, −2, 3),$ $v = 〈 1, 2, 3 〉$

Answer

x =

1 +

t,

y =

−2 +

2 t,

z =

3 +

3 t,

t \in ℝ;

\frac{x - 1}{1} =

\frac{y + 2}{2} =

\frac{z - 3}{3};

(0, −4, 0)

2-249 §2.5 source ↗

For the following exercises, point $P$ and vector $v$ are given. Let $L$ be the line passing through point $P$ with direction $v .$

$P (3, 1, 5),$ $v = \vec{Q R},$ where $Q (2, 2, 3)$ and $R (3, 2, 3)$

Answer

x =

3 +

t,

y =

1,

z =

5,

t \in ℝ;

y = 1, z = 5;

c. The line does not intersect the xy-plane.

2-251 §2.5 source ↗

For the following exercises, line $L$ is given.

$x =$ $1 +$ $t,$ $y =$ $3 +$ $t,$ $z =$ $5 +$ $4 t,$ $t \in ℝ$

Answer

P (1, 3, 5),

v = 〈 1, 1, 4 〉;

\sqrt{3}

2-253 §2.5 source ↗

For the following exercises, line $L$ is given.

Find the distance between point $A (−3, 1, 1)$ and the line of symmetric equations

$x = − y = − z .$

Answer

\frac{2 \sqrt{2}}{\sqrt{3}}

2-255 §2.5 source ↗

For the following exercises, lines $L_{1}$ and $L_{2}$ are given.

$L_{1} : x =$ $1 +$ $t,$ $y =$ $t,$ $z =$ $2 +$ $t,$ $t \in ℝ,$ $L_{2} : x -$ $3 =$ $y -$ $1 =$ $z -$ $3$

Answer

a. Parallel; b.

\frac{\sqrt{2}}{\sqrt{3}}

2-259 §2.5 source ↗

For the following exercises, lines $L_{1}$ and $L_{2}$ are given.

Find the point of intersection of the lines of equations $x = −2 y = 3 z$ and $x =$ $−5 -$ $t,$ $y =$ $−1 +$ $t,$ $z =$ $t -$ $11,$ $t \in ℝ .$

Answer

(−12, 6, −4)

2-261 §2.5 source ↗

For the following exercises, lines $L_{1}$ and $L_{2}$ are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting.

$L_{1} : x = y - 1 = − z$ and $L_{2} : x -$ $2 =$ $− y =$ $\frac{z}{2}$

Answer

The lines are skew.

2-263 §2.5 source ↗

For the following exercises, lines $L_{1}$ and $L_{2}$ are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting.

$L_{1} : x =$ $−1 +$ $2 t,$ $y =$ $1 +$ $3 t,$ $z =$ $7 t,$ $t \in ℝ$ and $L_{2} : x -$ $1 =$ $\frac{2}{3} (y - 4) =$ $\frac{2}{7} z -$ $2$

Answer

The lines are equal.

2-265 §2.5 source ↗

For the following exercises, lines $L_{1}$ and $L_{2}$ are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting.

Consider line $L$ of symmetric equations $x - 2 = − y = \frac{z}{2}$ and point $A (1, 1, 1) .$

a. Find parametric equations for a line parallel to $L$ that passes through point $A .$

b. Find symmetric equations of a line skew to $L$ and that passes through point $A .$

c. Find symmetric equations of a line that intersects $L$ and passes through point $A .$

Answer

x =

1 +

t,

y =

1 -

t,

z =

1 +

2 t,

t \in ℝ;

b. For instance, the line passing through

A

with direction vector

j : x = 1, z = 1;

c. For instance, the line passing through

A

and point

(2, 0, 0)

that belongs to

L

is a line that intersects;

L : \frac{x - 1}{−1} =

y -

1 =

z -

1

2-267 §2.5 source ↗

For the following exercises, point $P$ and vector $n$ are given.

$P (0, 0, 0),$ $n = 3 i - 2 j + 4 k$

Answer

3 x - 2 y + 4 z = 0;

3 x - 2 y + 4 z = 0

2-269 §2.5 source ↗

For the following exercises, point $P$ and vector $n$ are given.

$P (1, 2, 3),$ $n = 〈 1, 2, 3 〉$

Answer

(x - 1) +

2 (y - 2) +

3 (z - 3) =

0;

x +

2 y +

3 z -

14 =

0

2-271 §2.5 source ↗

For the following exercises, the equation of a plane is given.

[T] $4 x +$ $5 y +$ $10 z -$ $20 =$ $0$

Answer

a. $n =$ $4 i +$ $5 j +$ $10 k;$ b. $(5, 0, 0),$ $(0, 4, 0),$ and $(0, 0, 2);$
c.

This figure is the first octant of the 3-dimensional coordinate system. It has a triangle drawn with vertices on the x, y, and z axes.

2-273 §2.5 source ↗

For the following exercises, the equation of a plane is given.

$3 x - 2 y + 4 z = 0$

Answer

a. $n = 3 i - 2 j + 4 k;$ b. $(0, 0, 0);$
c.

This figure is the 3-dimensional coordinate system represented in a box. It has a tilted parallelogram inside the box representing a plane.

2-275 §2.5 source ↗

For the following exercises, the equation of a plane is given.

Given point $P (1, 2, 3)$ and vector $n = i + j,$ find point $Q$ on the x-axis such that $\vec{P Q}$ and $n$ are orthogonal.

Answer

(3, 0, 0)

2-277 §2.5 source ↗

For the following exercises, the equation of a plane is given.

Find parametric equations of the line passing through point $P (−2, 1, 3)$ that is perpendicular to the plane of equation $2 x - 3 y + z = 7 .$

Answer

x =

−2 +

2 t,

y =

1 -

3 t,

z =

3 +

t,

t \in ℝ

2-281 §2.5 source ↗

For the following exercises, points $P, Q, and R$ are given.

$P (1, 1, 1),$ $Q (2, 4, 3),$ and $R (−1, −2, −1)$

Answer

a. $−2 y + 3 z - 1 = 0;$ b. $〈 0,$ $−2,$ $3 〉 \cdot 〈 x -$ $1,$ $y -$ $1,$ $z -$ $1 〉 =$ $0;$ c. $x =$ $0,$ $y =$ $−2 t,$ $z =$ $3 t,$ $t \in ℝ$

Answers may vary by a sign, depending on how the vector cross multiplication is performed.

2-283 §2.5 source ↗

For the following exercises, points $P, Q, and R$ are given.

Consider the planes of equations $x + y + z = 1$ and $x + z = 0 .$

a. Show that the planes intersect.

b. Find symmetric equations of the line passing through point $P (1, 4, 6)$ that is parallel to the line of intersection of the planes.

Answer

a. Answers may vary; b.

\frac{x - 1}{1} =

\frac{z - 6}{−1},

y =

4

2-285 §2.5 source ↗

For the following exercises, points $P, Q, and R$ are given.

Find the scalar equation of the plane that passes through point $P (−1, 2, 1)$ and is perpendicular to the line of intersection of planes $x + y - z - 2 = 0$ and $2 x -$ $y +$ $3 z -$ $1 =$ $0 .$

Answer

2 x -

5 y -

3 z +

15 =

0

2-287 §2.5 source ↗

For the following exercises, points $P, Q, and R$ are given.

Determine whether the line of parametric equations $x =$ $1 +$ $2 t,$ $y =$ $−2 t,$ $z =$ $2 +$ $t,$ $t \in ℝ$ intersects the plane with equation $3 x +$ $4 y +$ $6 z -$ $7 =$ $0 .$ If it does intersect, find the point of intersection.

Answer

The line intersects the plane at point

P (−3, 4, 0) .

2-289 §2.5 source ↗

For the following exercises, points $P, Q, and R$ are given.

Find the distance from point $P (1, 5, −4)$ to the plane of equation $3 x -$ $y +$ $2 z -$ $6 =$ $0 .$

Answer

\frac{16}{\sqrt{14}}

2-291 §2.5 source ↗

For the following exercises, the equations of two planes are given.

[T] $x + y + z = 0,$ $2 x - y + z - 7 = 0$

Answer

a. The planes are neither parallel nor orthogonal; b.

62 °

2-293 §2.5 source ↗

For the following exercises, the equations of two planes are given.

$x - 5 y - z = 1,$ $5 x -$ $25 y -$ $5 z =$ $−3$

Answer

a. The planes are parallel.

2-295 §2.5 source ↗

For the following exercises, the equations of two planes are given.

Show that the lines of equations $x =$ $t,$ $y =$ $1 +$ $t,$ $z =$ $2 +$ $t,$ $t \in ℝ,$ and $\frac{x}{2} = \frac{y - 1}{3} = z - 3$ are skew, and find the distance between them.

Answer

\frac{1}{\sqrt{6}}

2-297 §2.5 source ↗

For the following exercises, the equations of two planes are given.

Consider point $C (−3, 2, 4)$ and the plane of equation $2 x + 4 y - 3 z = 8 .$

a. Find the radius of the sphere with center $C$ tangent to the given plane.

b. Find point P of tangency.

Answer

\frac{18}{\sqrt{29}};

P (- \frac{51}{29}, \frac{130}{29}, \frac{62}{29})

2-299 §2.5 source ↗

For the following exercises, the equations of two planes are given.

Two children are playing with a ball. The girl throws the ball to the boy. The ball travels in

the air, curves $3$ ft to the right, and falls $5$ ft away from the girl (see the following figure). If the plane that contains the trajectory of the ball is perpendicular to the ground, find its equation.

This figure is the image of two children throwing a ball. The path of the ball is represented with an arc. The distance from the child throwing the ball to the point where the ball hits is 5 feet. The distance from the second child to where the ball hits is 3 feet.

Answer

4 x - 3 y = 0

2-301 §2.5 source ↗

For the following exercises, the equations of two planes are given.

[T] Consider $r (t) =$ $〈 sin t,$ $cos t,$ $2 t 〉$ the position vector of a particle at time $t \in [0, 3],$ where the components of r are expressed in centimeters and time is measured in seconds. Let $\vec{O P}$ be the position vector of the particle after $1$ sec.

a. Determine the velocity vector $v (1)$ of the particle after $1$ sec.

b. Find the scalar equation of the plane that is perpendicular to $v (1)$ and passes through point $P .$ This plane is called the normal plane to the path of the particle at point $P .$

c. Use a CAS to visualize the path of the particle along with the velocity vector and normal plane at point $P .$

Answer

a. $v (1) =$ $〈 cos 1,$ $− sin 1,$ $2 〉;$ b. $(cos 1) (x - sin 1) -$ $(sin 1) (y - cos 1) +$ $2 (z - 2) =$ $0;$
c.

This figure is the first octant of the 3-dimensional coordinate system. It has a parallelogram grid drawn representing a plane. There is a curve from y = 1 increasing. The curve intersects the plane. At the point the curve intersects the plane, there is a vector labeled “v(1).” It is upward parallel to the z-axis.

Checkpoint 2.52 §2.6 source ↗

Sketch or use a graphing tool to view the graph of the cylindrical surface defined by equation

z = y^{2} .

Answer

This figure is a surface above the x y plane. A cross section of this surface parallel to the y z plane would be a parabola. The surface sits on top of the x y plane.

Checkpoint 2.53 §2.6 source ↗

A hyperboloid of one sheet is any surface that can be described with an equation of the form

\frac{x^{2}}{a^{2}} +

\frac{y^{2}}{b^{2}} -

\frac{z^{2}}{c^{2}} =

1 .

Describe the traces of the hyperboloid of one sheet given by equation

\frac{x^{2}}{3^{2}} +

\frac{y^{2}}{2^{2}} -

\frac{z^{2}}{5^{2}} =

1 .

Answer

The traces parallel to the xy-plane are ellipses and the traces parallel to the xz- and yz-planes are hyperbolas. Specifically, the trace in the xy-plane is ellipse $\frac{x^{2}}{3^{2}} +$ $\frac{y^{2}}{2^{2}} =$ $1,$ the trace in the xz-plane is hyperbola $\frac{x^{2}}{3^{2}} -$ $\frac{z^{2}}{5^{2}} =$ $1,$ and the trace in the yz-plane is hyperbola $\frac{y^{2}}{2^{2}} - \frac{z^{2}}{5^{2}} = 1$ (see the following figure).

This figure has four images. The first image is an ellipse centered at the origin of a rectangular coordinate system. It intersects the x axis at -3 and 3. It intersects the y axis at -2 and 2. The second image is the graph of a hyperbola. It is two curves one opening in the negative x direction and a symmetric one in the positive x direction. The third image is the graph of a hyperbola in the y z plane. It is opening in the negative y direction and a symmetric curve opening in the positive y direction. The fourth image is a 3-dimensional surface. It top and bottom cross sections would be circular. A vertical intersection would be a hyperbola.

Checkpoint 2.54 §2.6 source ↗

Identify the surface represented by equation

9 x^{2} +

y^{2} -

z^{2} +

2 z -

10 =

0 .

Answer

Hyperboloid of one sheet, centered at

(0, 0, 1)

2-303 §2.6 source ↗

For the following exercises, sketch and describe the cylindrical surface of the given equation.

[T] $x^{2} + z^{2} = 1$

Answer

The surface is a cylinder with the rulings parallel to the y-axis.

This figure is a circular cylinder inside of a box. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

2-305 §2.6 source ↗

For the following exercises, sketch and describe the cylindrical surface of the given equation.

[T] $z = cos (\frac{π}{2} + x)$

Answer

The surface is a cylinder with rulings parallel to the y-axis.

This figure is a surface inside of a box. Its cross section parallel to the x z plane would be a cosine curve. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

2-307 §2.6 source ↗

For the following exercises, sketch and describe the cylindrical surface of the given equation.

[T] $z = 9 - y^{2}$

Answer

The surface is a cylinder with rulings parallel to the x-axis.

This figure is a surface inside of a box. Its cross section parallel to the y z plane would be an upside down parabola. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

2-309 §2.6 source ↗

For the following exercises, the graph of a quadric surface is given.

Answer

a. Cylinder; b. The x-axis

2-311 §2.6 source ↗

For the following exercises, the graph of a quadric surface is given.

This figure is a surface in the 3-dimensional coordinate system. There are two conical shapes facing away from each other. They have the x axis through the center.

Answer

a. Hyperboloid of two sheets; b. The x-axis

2-313 §2.6 source ↗

For the following exercises, match the given quadric surface with its corresponding equation in standard form.

Hyperboloid of two sheets

Answer

2-315 §2.6 source ↗

For the following exercises, match the given quadric surface with its corresponding equation in standard form.

Elliptic paraboloid

Answer

2-317 §2.6 source ↗

For the following exercises, match the given quadric surface with its corresponding equation in standard form.

Hyperboloid of one sheet

Answer

2-319 §2.6 source ↗

For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface.

$− x^{2} +$ $36 y^{2} +$ $36 z^{2} =$ $9$

Answer

-

\frac{x^{2}}{9} +

\frac{y^{2}}{\frac{1}{4}} +

\frac{z^{2}}{\frac{1}{4}} =

1,

hyperboloid of one sheet with the x-axis as its axis of symmetry

2-321 §2.6 source ↗

For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface.

$−3 x^{2} +$ $5 y^{2} -$ $z^{2} =$ $10$

Answer

-

\frac{x^{2}}{\frac{10}{3}} +

\frac{y^{2}}{2} -

\frac{z^{2}}{10} =

1,

hyperboloid of two sheets with the y-axis as its axis of symmetry

2-323 §2.6 source ↗

For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface.

$5 y = x^{2} - z^{2}$

Answer

y = - \frac{z^{2}}{5} + \frac{x^{2}}{5},

hyperbolic paraboloid with the y-axis as its axis of symmetry

2-325 §2.6 source ↗

For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface.

$x^{2} +$ $5 y^{2} +$ $3 z^{2} -$ $15 =$ $0$

Answer

\frac{x^{2}}{15} +

\frac{y^{2}}{3} +

\frac{z^{2}}{5} =

1,

ellipsoid

2-327 §2.6 source ↗

For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface.

$x^{2} +$ $5 y^{2} -$ $8 z^{2} =$ $0$

Answer

\frac{x^{2}}{40} +

\frac{y^{2}}{8} -

\frac{z^{2}}{5} =

0,

elliptic cone with the z-axis as its axis of symmetry

2-329 §2.6 source ↗

For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface.

$6 x = 3 y^{2} + 2 z^{2}$

Answer

x = \frac{y^{2}}{2} + \frac{z^{2}}{3},

elliptic paraboloid with the x-axis as its axis of symmetry

2-331 §2.6 source ↗

For the following exercises, find the trace of the given quadric surface in the specified plane of coordinates and sketch it.

[T] $x^{2} +$ $z^{2} +$ $4 y =$ $0,$ $z =$ $0$

Answer

Parabola $y = - \frac{x^{2}}{4},$

This figure is the graph of an upside down parabola with its highest point at the origin of a rectangular coordinate system.

2-333 §2.6 source ↗

For the following exercises, find the trace of the given quadric surface in the specified plane of coordinates and sketch it.

[T] $−4 x^{2} +$ $25 y^{2} +$ $z^{2} =$ $100,$ $x =$ $0$

Answer

Ellipse $\frac{y^{2}}{4} +$ $\frac{z^{2}}{100} =$ $1,$

This figure is the graph of an ellipse centered at the origin of a rectangular coordinate system.

2-335 §2.6 source ↗

For the following exercises, find the trace of the given quadric surface in the specified plane of coordinates and sketch it.

[T] $x^{2} +$ $\frac{y^{2}}{4} +$ $\frac{z^{2}}{100} =$ $1,$ $x =$ $0$

Answer

Ellipse $\frac{y^{2}}{4} +$ $\frac{z^{2}}{100} =$ $1,$

2-337 §2.6 source ↗

For the following exercises, find the trace of the given quadric surface in the specified plane of coordinates and sketch it.

Use the graph of the given quadric surface to answer the questions.

This figure is a surface inside of a box. It is an oval solid on its side. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

a. Specify the name of the quadric surface.

b. Which of the equations— $16 x^{2} +$ $9 y^{2} +$ $36 z^{2} =$ $3600,$ $9 x^{2} +$ $36 y^{2} +$ $16 z^{2} =$ $3600,$ or $36 x^{2} +$ $9 y^{2} +$ $16 z^{2} =$ $3600$ —corresponds to the graph?

c. Use b. to write the equation of the quadric surface in standard form.

Answer

a. Ellipsoid; b. The third equation; c.

\frac{x^{2}}{100} +

\frac{y^{2}}{400} +

\frac{z^{2}}{225} =

1

2-339 §2.6 source ↗

For the following exercises, the equation of a quadric surface is given.

$x^{2} +$ $2 z^{2} +$ $6 x -$ $8 z +$ $1 =$ $0$

Answer

\frac{{(x + 3)}^{2}}{16} +

\frac{{(z - 2)}^{2}}{8} =

1;

b. Cylinder centered at

(−3, 2)

with rulings parallel to the y-axis

2-341 §2.6 source ↗

For the following exercises, the equation of a quadric surface is given.

$x^{2} +$ $4 y^{2} -$ $4 z^{2} -$ $6 x -$ $16 y -$ $16 z +$ $5 =$ $0$

Answer

\frac{{(x - 3)}^{2}}{4} +

{(y - 2)}^{2} -

{(z + 2)}^{2} =

1;

b. Hyperboloid of one sheet centered at

(3, 2, −2),

with the z-axis as its axis of symmetry

2-343 §2.6 source ↗

For the following exercises, the equation of a quadric surface is given.

$x^{2} +$ $\frac{y^{2}}{4} -$ $\frac{z^{2}}{3} +$ $6 x +$ $9 =$ $0$

Answer

{(x + 3)}^{2} +

\frac{y^{2}}{4} -

\frac{z^{2}}{3} =

0;

b. Elliptic cone centered at

(−3, 0, 0),

with the z-axis as its axis of symmetry

2-345 §2.6 source ↗

For the following exercises, the equation of a quadric surface is given.

Write the standard form of the equation of the ellipsoid centered at the origin that passes through points $A (2, 0, 0),$ $B (0, 0, 1),$ and $C (\frac{1}{2}, \sqrt{11}, \frac{1}{2}) .$

Answer

\frac{x^{2}}{4} +

\frac{y^{2}}{16} +

z^{2} =

1

2-347 §2.6 source ↗

For the following exercises, the equation of a quadric surface is given.

Determine the intersection points of elliptic cone $x^{2} - y^{2} - z^{2} = 0$ with the line of symmetric equations $\frac{x - 1}{2} =$ $\frac{y + 1}{3} =$ $z .$

Answer

(1, −1, 0)

and

(\frac{13}{3}, 4, \frac{5}{3})

2-349 §2.6 source ↗

For the following exercises, the equation of a quadric surface is given.

Find an equation of the quadric surface with points $P (x, y, z)$ that are equidistant from point $Q (0, −1, 0)$ and plane of equation $y = 1 .$ Identify the surface.

Answer

x^{2} + z^{2} + 4 y = 0,

elliptic paraboloid

2-351 §2.6 source ↗

For the following exercises, the equation of a quadric surface is given.

If the surface of a parabolic reflector is described by equation $400 z = x^{2} + y^{2},$ find the focal point of the reflector.

Answer

(0, 0, 100)

2-355 §2.6 source ↗

For the following exercises, the equation of a quadric surface is given.

[T] The intersection between cylinder ${(x - 1)}^{2} + y^{2} = 1$ and sphere $x^{2} + y^{2} + z^{2} = 4$ is called a Viviani curve.

This figure is a surface inside of a box. It is a sphere with a right circular cylinder through the sphere vertically. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

a. Solve the system consisting of the equations of the surfaces to find the equations of the intersection curve. (Hint: Find $x$ and $y$ in terms of $z .)$

b. Use a computer algebra system (CAS) to visualize the intersection curve on sphere $x^{2} + y^{2} + z^{2} = 4 .$

Answer

a. $x =$ $2 -$ $\frac{z^{2}}{2},$ $y =$ $\pm \frac{z}{2} \sqrt{4 - z^{2}},$ where $z \in [−2, 2];$
b.

This figure is a surface inside of a box. It is a sphere with a figure eight curve on the side of the sphere. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

2-357 §2.6 source ↗

For the following exercises, the equation of a quadric surface is given.

[T] Use a CAS to create the intersection between cylinder $9 x^{2} + 4 y^{2} = 18$ and ellipsoid $36 x^{2} +$ $16 y^{2} +$ $9 z^{2} =$ $144,$ and find the equations of the intersection curves.

Answer

This figure is a surface inside of a box. It is a solid oval with an elliptical cylinder vertically intersecting. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

two ellipses of equations $\frac{x^{2}}{2} + \frac{y^{2}}{\frac{9}{2}} = 1$ in planes $z = ± 2 \sqrt{2}$

2-359 §2.6 source ↗

For the following exercises, the equation of a quadric surface is given.

[T] In cartography, Earth is approximated by an oblate spheroid rather than a sphere. The radii at the equator and poles are approximately $3963$ mi and $3950$ mi, respectively.

a. Write the equation in standard form of the ellipsoid that represents the shape of Earth. Assume the center of Earth is at the origin and that the trace formed by plane $z = 0$ corresponds to the equator.

b. Sketch the graph.

c. Find an equation of the intersection curve of the surface with plane $z = 1000$ that is parallel to the xy-plane. The intersection curve is called a parallel.

d. Find an equation of the intersection curve of the surface with plane $x + y = 0$ that passes through the z-axis. The intersection curve is called a meridian.

Answer

a. $\frac{x^{2}}{3963^{2}} +$ $\frac{y^{2}}{3963^{2}} +$ $\frac{z^{2}}{3950^{2}} =$ $1;$
b.

This figure is a surface inside of a box. It is a sphere. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

;
c. The intersection curve is the ellipse of equation $\frac{x^{2}}{3963^{2}} +$ $\frac{y^{2}}{3963^{2}} =$ $\frac{(2950) (4950)}{3950^{2}},$ and the intersection is an ellipse.; d. The intersection curve is the ellipse of equation $\frac{2 y^{2}}{3963^{2}} +$ $\frac{z^{2}}{3950^{2}} =$ $1 .$

2-361 §2.6 source ↗

For the following exercises, the equation of a quadric surface is given.

[T] A heart-shaped surface is given by equation ${(x^{2} + \frac{9}{4} y^{2} + z^{2} - 1)}^{3} -$ $x^{2} z^{3} -$ $\frac{9}{80} y^{2} z^{3} =$ $0 .$

a. Use a CAS to graph the surface that models this shape.

b. Determine and sketch the trace of the heart-shaped surface on the xz-plane.

Answer

This figure is a surface inside of a box. It is a heart. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

b. The intersection curve is ${(x^{2} + z^{2} - 1)}^{3} -$ $x^{2} z^{3} =$ $0 .$

This figure is a curve on a rectangular coordinate system. It is the shape of a heart centered about the y-axis.

Checkpoint 2.55 §2.7 source ↗

Point

R

has cylindrical coordinates

(5, \frac{π}{6}, 4)

. Plot

R

and describe its location in space using rectangular, or Cartesian, coordinates.

Answer

The rectangular coordinates of the point are $(\frac{5 \sqrt{3}}{2}, \frac{5}{2}, 4) .$

This figure is the 3-dimensional coordinate system. There is a point labeled “(5, pi/6, 4).” The point is located above a line segment in the x y-plane labeled r = 5 that is pi/6 degrees from the x-axis. The distance from the x y-plane to the point is labeled “z = 4.”

Checkpoint 2.56 §2.7 source ↗

Convert point

(−8, 8, −7)

from Cartesian coordinates to cylindrical coordinates.

Answer

(8 \sqrt{2}, \frac{3 π}{4}, −7)

Checkpoint 2.57 §2.7 source ↗

Describe the surface with cylindrical equation

r = 6 .

Answer

This surface is a cylinder with radius $6 .$

This figure is a right circular cylinder. It is upright with the z-axis through the center. It is on top of the x y plane.

Checkpoint 2.58 §2.7 source ↗

Plot the point with spherical coordinates

(2, - \frac{5 π}{6}, \frac{π}{6})

and describe its location in both rectangular and cylindrical coordinates.

Answer

This figure is of the 3-dimensional coordinate system. It has a point. There is a line segment from the origin to the point. The angle between this line segment and the z-axis is phi. There is a line segment in the x y-plane from the origin to the shadow of the point.The angle between the x-axis and rho is theta.

Cartesian: $(- \frac{\sqrt{3}}{2}, - \frac{1}{2}, \sqrt{3}),$ cylindrical: $(1, - \frac{5 π}{6}, \sqrt{3})$

Checkpoint 2.59 §2.7 source ↗

Describe the surfaces defined by the following equations.

a. $ρ = 13$

b. $θ = \frac{2 π}{3}$

c. $φ = \frac{π}{4}$

Answer

a. This is the set of all points

13

units from the origin. This set forms a sphere with radius

13 .

b. This set of points forms a half plane. The angle between the half plane and the positive x-axis is

θ = \frac{2 π}{3} .

c. Let

P

be a point on this surface. The position vector of this point forms an angle of

φ = \frac{π}{4}

with the positive z-axis, which means that points closer to the origin are closer to the axis. These points form a half-cone.

Checkpoint 2.60 §2.7 source ↗

Sydney, Australia is at

34 ° S

and

151 ° E .

Express Sydney’s location in spherical coordinates.

Answer

(4000, 151 °, 124 °)

Checkpoint 2.61 §2.7 source ↗

Which coordinate system is most appropriate for creating a star map, as viewed from Earth (see the following figure)?

This figure is a circle with a star chart in the middle.

How should we orient the coordinate axes?

Answer

Spherical coordinates with the origin located at the center of the earth, the z-axis aligned with the North Pole, and the x-axis aligned with the prime meridian

2-363 §2.7 source ↗

Use the following figure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems.

This figure is the first octant of the 3-dimensional coordinate system. There is a line segment from the origin upwards. It is labeled “rho.” The angle between this line segment and the z-axis is labeled “phi.” There is also a broken line from the origin to the shadow of the point. This line segment is in the x y-plane and is labeled “r.” The angle between r and the x-axis is labeled “theta.”

For the following exercises, the cylindrical coordinates $(r, θ, z)$ of a point are given. Find the rectangular coordinates $(x, y, z)$ of the point.

$(4, \frac{π}{6}, 3)$

Answer

(2 \sqrt{3}, 2, 3)

2-365 §2.7 source ↗

Use the following figure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems.

For the following exercises, the cylindrical coordinates $(r, θ, z)$ of a point are given. Find the rectangular coordinates $(x, y, z)$ of the point.

$(4, \frac{7 π}{6}, 3)$

Answer

(−2 \sqrt{3}, −2, 3)

2-367 §2.7 source ↗

For the following exercises, the rectangular coordinates $(x, y, z)$ of a point are given. Find the cylindrical coordinates $(r, θ, z)$ of the point.

$(1, \sqrt{3}, 2)$

Answer

(2, \frac{π}{3}, 2)

2-369 §2.7 source ↗

For the following exercises, the rectangular coordinates $(x, y, z)$ of a point are given. Find the cylindrical coordinates $(r, θ, z)$ of the point.

$(3, −3, 7)$

Answer

(3 \sqrt{2}, - \frac{π}{4}, 7)

2-371 §2.7 source ↗

For the following exercises, the equation of a surface in cylindrical coordinates is given.

Find an equation of the surface in rectangular coordinates. Identify and graph the surface.

[T] $r = 4$

Answer

A cylinder of equation $x^{2} + y^{2} = 16,$ with its center at the origin and rulings parallel to the z-axis,

This figure is a right circular cylinder, vertical. It is inside of a box. The edges of the box represent the x, y, and z axes.

2-373 §2.7 source ↗

For the following exercises, the equation of a surface in cylindrical coordinates is given.

Find an equation of the surface in rectangular coordinates. Identify and graph the surface.

[T] $r^{2} cos (2 θ) +$ $z^{2} +$ $1 =$ $0$

Answer

Hyperboloid of two sheets of equation $− x^{2} +$ $y^{2} -$ $z^{2} =$ $1,$ with the y-axis as the axis of symmetry,

This figure is a elliptic cone surface that is horizontal. It is inside of a box. The edges of the box represent the x, y, and z axes.

2-375 §2.7 source ↗

For the following exercises, the equation of a surface in cylindrical coordinates is given.

Find an equation of the surface in rectangular coordinates. Identify and graph the surface.

[T] $r = 2 cos θ$

Answer

Cylinder of equation $x^{2} - 2 x + y^{2} = 0,$ with a center at $(1, 0, 0)$ and radius $1,$ with rulings parallel to the z-axis,

2-377 §2.7 source ↗

For the following exercises, the equation of a surface in cylindrical coordinates is given.

Find an equation of the surface in rectangular coordinates. Identify and graph the surface.

[T] $r = 2 sec θ$

Answer

Plane of equation $x = 2,$

This figure is a vertical parallelogram where x = 2 and parallel to the y z-plane. It is inside of a box. The edges of the box represent the x, y, and z axes.

2-379 §2.7 source ↗

For the following exercises, the equation of a surface in rectangular coordinates is given. Find an equation of the surface in cylindrical coordinates.

$z = 3$

Answer

z = 3

2-381 §2.7 source ↗

For the following exercises, the equation of a surface in rectangular coordinates is given. Find an equation of the surface in cylindrical coordinates.

$x^{2} + y^{2} + z^{2} = 9$

Answer

r^{2} + z^{2} = 9

2-383 §2.7 source ↗

For the following exercises, the equation of a surface in rectangular coordinates is given. Find an equation of the surface in cylindrical coordinates.

$x^{2} + y^{2} - 16 x = 0$

Answer

r =

16 cos θ,

r =

0

2-385 §2.7 source ↗

For the following exercises, the spherical coordinates $(ρ, θ, φ)$ of a point are given. Find the rectangular coordinates $(x, y, z)$ of the point.

$(3, 0, π)$

Answer

(0, 0, −3)

2-387 §2.7 source ↗

For the following exercises, the spherical coordinates $(ρ, θ, φ)$ of a point are given. Find the rectangular coordinates $(x, y, z)$ of the point.

$(12, - \frac{π}{4}, \frac{π}{4})$

Answer

(6, −6, 6 \sqrt{2})

2-389 §2.7 source ↗

For the following exercises, the rectangular coordinates $(x, y, z)$ of a point are given. Find the spherical coordinates $(ρ, θ, φ)$ of the point. Express the measure of the angles in degrees rounded to the nearest integer.

$(4, 0, 0)$

Answer

(4, 0, 90 °)

2-391 §2.7 source ↗

$(0, 3, 0)$

Answer

(3, 90 °, 90 °)

2-393 §2.7 source ↗

For the following exercises, the equation of a surface in spherical coordinates is given. Find an equation of the surface in rectangular coordinates. Identify and graph the surface.

[T] $ρ = 3$

Answer

Sphere of equation $x^{2} + y^{2} + z^{2} = 9$ centered at the origin with radius $3,$

This figure is a sphere. It is inside of a box. The edges of the box represent the x, y, and z axes.

2-395 §2.7 source ↗

For the following exercises, the equation of a surface in spherical coordinates is given. Find an equation of the surface in rectangular coordinates. Identify and graph the surface.

[T] $ρ = 2 cos φ$

Answer

Sphere of equation $x^{2} +$ $y^{2} +$ ${(z - 1)}^{2} =$ $1$ centered at $(0, 0, 1)$ with radius $1,$

This figure is a sphere of radius 1 centered in a box. The center of the sphere is the point (0, 0, 1).

2-397 §2.7 source ↗

For the following exercises, the equation of a surface in spherical coordinates is given. Find an equation of the surface in rectangular coordinates. Identify and graph the surface.

[T] $φ = \frac{π}{2}$

Answer

The xy-plane of equation $z = 0,$

This figure is a parallelogram representing a plane. It is parallel to the x y-plane at z = 0. It is inside of a box. The edges of the box represent the x, y, and z axes.

2-399 §2.7 source ↗

For the following exercises, the equation of a surface in rectangular coordinates is given. Find an equation of the surface in spherical coordinates. Identify the surface.

$x^{2} +$ $y^{2} -$ $3 z^{2} =$ $0,$ $z \neq 0$

Answer

φ = \frac{π}{3}

φ = \frac{2 π}{3};

Elliptic cone

2-401 §2.7 source ↗

For the following exercises, the equation of a surface in rectangular coordinates is given. Find an equation of the surface in spherical coordinates. Identify the surface.

$z = 6$

Answer

ρ cos φ = 6;

Plane at

z = 6

2-403 §2.7 source ↗

For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle $φ$ in radians rounded to four decimal places.

[T] $(1, \frac{π}{4}, 3)$

Answer

(\sqrt{10}, \frac{π}{4}, 0.3218)

2-405 §2.7 source ↗

For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle $φ$ in radians rounded to four decimal places.

$(3, \frac{π}{2}, 3)$

Answer

(3 \sqrt{2}, \frac{π}{2}, \frac{π}{4})

2-407 §2.7 source ↗

For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates.

$(2, - \frac{π}{4}, \frac{π}{2})$

Answer

(2, - \frac{π}{4}, 0)

2-409 §2.7 source ↗

For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates.

$(8, \frac{π}{3}, \frac{π}{2})$

Answer

(8, \frac{π}{3}, 0)

2-411 §2.7 source ↗

For the following exercises, find the most suitable system of coordinates to describe the solids.

The solid situated in the first octant with a vertex at the origin and enclosed by a cube of edge length $a,$ where $a > 0$

Answer

Cartesian system,

{(x, y, z) | 0 \leq x \leq a, 0 \leq y \leq a, 0 \leq z \leq a}

2-413 §2.7 source ↗

For the following exercises, find the most suitable system of coordinates to describe the solids.

A solid inside sphere $x^{2} + y^{2} + z^{2} = 9$ and outside cylinder ${(x - \frac{3}{2})}^{2} +$ $y^{2} =$ $\frac{9}{4}$

Answer

Cylindrical system,

{\begin{cases} (r, θ, z) | r^{2} + z^{2} \leq 9, r \geq 0, \frac{π}{2} \leq θ \leq \frac{3 π}{2} \\ , (r \geq 3 \cos θ, - \frac{π}{2} \leq θ \leq \frac{π}{2}) \end{cases}}

2-415 §2.7 source ↗

For the following exercises, find the most suitable system of coordinates to describe the solids.

[T] Use a CAS to graph the region between elliptic paraboloid $z = x^{2} + y^{2}$ and cone $x^{2} + y^{2} - z^{2} = 0 .$ Then describe the region in cylindrical coordinates.

Answer

The region is described by the set of points ${(r, θ, z) | 0 \leq r \leq 1, 0 \leq θ \leq 2 π, r^{2} \leq z \leq r} .$

This figure is a paraboloid, vertical. It is inside of a box. The edges of the box represent the x, y, and z axes.

2-417 §2.7 source ↗

For the following exercises, find the most suitable system of coordinates to describe the solids.

Washington, DC, is located at $39 °$ N and $77 °$ W (see the following figure). Assume the radius of Earth is $4000$ mi. Express the location of Washington, DC, in spherical coordinates.

This figure is an image of a globe. On the globe there is a point labeled where Washington, DC is located. It is labeled with 39 degrees north latitude and 77 degrees west longitude.

Answer

(4000, − 77 °, 51 °)

2-419 §2.7 source ↗

For the following exercises, find the most suitable system of coordinates to describe the solids.

Find the latitude and longitude of Rio de Janeiro if its spherical coordinates are $(4000, − 43.17 °, 102.91 °) .$

Answer

43.17 ° W,

22.91 ° S

2-421 §2.7 source ↗

For the following exercises, find the most suitable system of coordinates to describe the solids.

[T] Consider the torus of equation ${(x^{2} + y^{2} + z^{2} + R^{2} - r^{2})}^{2} =$ $4 R^{2} (x^{2} + y^{2}),$ where $R \geq r > 0 .$

a. Write the equation of the torus in spherical coordinates.

b. If $R = r,$ the surface is called a horn torus. Show that the equation of a horn torus in spherical coordinates is $ρ = 2 R sin φ .$

c. Use a CAS to graph the horn torus with $R = r = 2$ in spherical coordinates.

Answer

a. $ρ^{2} = 0,$ $ρ +$ $R^{2} -$ $r^{2} -$ $2 R sin φ =$ $0;$
c.

This figure is a torus. It is inside of a box. The edges of the box represent the x, y, and z axes.

2-423 §2.R source ↗

For the following exercises, determine whether the statement is true or false. Justify the answer with a proof or a counterexample.

For vectors $a$ and $b$ and any given scalar $c,$ $c (a \cdot b) =$ $(c a) \cdot b .$

Answer

True

2-425 §2.R source ↗

For the following exercises, determine whether the statement is true or false. Justify the answer with a proof or a counterexample.

The symmetric equation for the line of intersection between two planes $x + y + z = 2$ and $x + 2 y - 4 z = 5$ is given by $-$ $\frac{x - 1}{6} =$ $\frac{y - 1}{5} =$ $z .$

Answer

False

2-427 §2.R source ↗

For the following exercises, use the given vectors to find the quantities.

$a =$ $9 i -$ $2 j,$ $b =$ $−3 i +$ $j$

a. $3 a + b$

b. $‖ a ‖$

c. $a \times ‖ b \times c ‖$

d. $b \cdot a$

Answer

〈 24, −5 〉;

\sqrt{85};

c. Can’t cross a vector with a scalar; d.

−29

2-429 §2.R source ↗

For the following exercises, use the given vectors to find the quantities.

Find the values of $a$ such that vectors $〈 2, 4, a 〉$ and $〈 0, −1, a 〉$ are orthogonal.

Answer

a = ± 2

2-431 §2.R source ↗

For the following exercises, find the unit vectors.

Find the unit vector that has the same direction as vector $v$ that begins at $(1, 4, 10)$ and ends at $(3, 0, 4) .$

Answer

〈 \frac{1}{\sqrt{14}},

-

\frac{2}{\sqrt{14}},

-

\frac{3}{\sqrt{14}} 〉

2-433 §2.R source ↗

For the following exercises, find the area or volume of the given shapes.

The parallelepiped formed by $a =$ $〈 1,$ $4,$ $1 〉 and b =$ $〈 3,$ $6,$ $2 〉,$ and $c = 〈 −2, 1, −5 〉$

Answer

27

2-435 §2.R source ↗

For the following exercises, find the vector and parametric equations of the line with the given properties.

The line that passes through points $(1, 3, 5)$ and $(−2, 6, −3)$

Answer

x =

1 -

3 t,

y =

3 +

3 t,

z =

5 -

8 t,

r (t) =

(1 - 3 t) i +

3 (1 + t) j +

(5 - 8 t) k

2-437 §2.R source ↗

For the following exercises, find an equation of the plane with the given properties.

The plane that passes through points $(0, 1, 5),$ $(2, −1, 6),$ $and (3, 2, 5) .$

Answer

− x + 3 y + 8 z = 43

2-439 §2.R source ↗

For the following exercises, find the traces for the surfaces in planes $x =$ $k,$ $y =$ $k,$ $and z =$ $k .$ Then, describe and draw the surfaces.

$x^{2} = y^{2} + z^{2}$

Answer

$x = k$ trace: $k^{2} = y^{2} + z^{2}$ is a circle, $y = k$ trace: $x^{2} - z^{2} = k^{2}$ is a hyperbola (or a pair of lines if $k = 0),$ $z = k$ trace: $x^{2} - y^{2} = k^{2}$ is a hyperbola (or a pair of lines if $k = 0) .$ The surface is a cone.

This figure is an elliptical cone on its side. It is inside of a box. The edges of the box represent the x, y, and z axes.

2-441 §2.R source ↗

For the following exercises, write the given equation in cylindrical coordinates and spherical coordinates.

$z = x^{2} + y^{2} - 1$

Answer

Cylindrical:

z = r^{2} - 1,

spherical:

cos φ =

ρ {sin}^{2} φ -

\frac{1}{ρ}

2-443 §2.R source ↗

For the following exercises, convert the given equations from cylindrical or spherical coordinates to rectangular coordinates. Identify the given surface.

$r^{2} -$ $2 r cos (θ) +$ $z^{2} =$ $1$

Answer

x^{2} -

2 x +

y^{2} +

z^{2} =

1,

sphere

2-445 §2.R source ↗

For the following exercises, consider a small boat crossing a river.

When the boat reaches the shore, two ropes are thrown to people to help pull the boat ashore. One rope is at an angle of $25 °$ and the other is at $35 ° .$ If the boat must be pulled straight and at a force of $500 N,$ find the magnitude of force for each rope (see the following figure).

This figure is overtop of a boat. From the front of the boat is a horizontal vector. It is labeled “500 N.” There are two other line segments from the boat. The first one forms an angle with the horizontal vector of 35 degrees above the vector. The second line segment forms an angle of 25 degrees below the vector.

Answer

331 N, and 244 N

2-447 §2.R source ↗

For the following exercises, consider a small boat crossing a river.

Calculate the work done by moving a particle from position $(1, 2, 0)$ to $(8, 4, 5)$ along a straight line with a force $F = 2 i + 3 j - k .$

Answer

15 J

2-449 §2.R source ↗

The following problems consider your unsuccessful attempt to take the tire off your car using a wrench to loosen the bolts. Assume the wrench is $0.3$ m long and you are able to apply a 200-N force.

Someone lends you a tire jack and you are now able to apply a 200-N force at an $80 °$ angle. Is your resulting torque going to be more or less? What is the new resulting torque at the center of the bolt? Assume this force is not enough to loosen the bolt.

Answer

More,

59.09

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Problems and answers are adapted from Calculus Volume 3 by Gilbert Strang and Edwin “Jed” Herman, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/calculus-volume-3/pages/1-introduction. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.