Chapter 12: Three DimensionsContent check — Calculus (Guichard), Chapter 12: Three Dimensions

Find the lengths of the sides of the triangle with vertices $(\mathrm{1,0,1})$⁠, $(\mathrm{2,2},-1)$⁠, and $(-\mathrm{3,2},-2)$⁠.

Find the lengths of the sides of the triangle with vertices $(\mathrm{2,2,3})$⁠, $(\mathrm{8,6,5})$⁠, and $(-\mathrm{1,0,2})$⁠. Why do the results tell you that this isn't really a triangle?

Find an equation of the sphere with center at $(\mathrm{1,1,1})$ and radius 2.

Find an equation of the sphere with center at $(2,-\mathrm{1,3})$ and radius 5.

Find an equation of the sphere with center $(3,-2,1)$ and that goes through the point $(4,2,5)$⁠.

Find an equation of the sphere with center at $(\mathrm{2,1},-1)$ and radius 4. Find an equation for the intersection of this sphere with the $y$⁠-⁠$z$ plane; describe this intersection geometrically.

Find $|𝐯|$⁠, $𝐯+𝐰$⁠, $𝐯-𝐰$⁠, $|𝐯+𝐰|$⁠, $|𝐯-𝐰|$ and $-2𝐯$ for $𝐯=⟨\mathrm{1,3}⟩$ and $𝐰=⟨-1,-5⟩$⁠.

Find $|𝐯|$⁠, $𝐯+𝐰$⁠, $𝐯-𝐰$⁠, $|𝐯+𝐰|$⁠, $|𝐯-𝐰|$ and $-2𝐯$ for $𝐯=⟨\mathrm{1,2,3}⟩$ and $𝐰=⟨-\mathrm{1,2},-3⟩$⁠.

Find $|𝐯|$⁠, $𝐯+𝐰$⁠, $𝐯-𝐰$⁠, $|𝐯+𝐰|$⁠, $|𝐯-𝐰|$ and $-2𝐯$ for $𝐯=⟨\mathrm{1,0,1}⟩$ and $𝐰=⟨-1,-\mathrm{2,2}⟩$⁠.

Find $|𝐯|$⁠, $𝐯+𝐰$⁠, $𝐯-𝐰$⁠, $|𝐯+𝐰|$⁠, $|𝐯-𝐰|$ and $-2𝐯$ for $𝐯=⟨1,-\mathrm{1,1}⟩$ and $𝐰=⟨\mathrm{0,0,3}⟩$⁠.

Find $|𝐯|$⁠, $𝐯+𝐰$⁠, $𝐯-𝐰$⁠, $|𝐯+𝐰|$⁠, $|𝐯-𝐰|$ and $-2𝐯$ for $𝐯=⟨\mathrm{3,2,1}⟩$ and $𝐰=⟨-1,-1,-1⟩$⁠.

Let $P=(\mathrm{4,5,6})$⁠, $Q=(\mathrm{1,2},-5)$⁠. Find ${\displaystyle \overrightarrow{PQ}}$⁠. Find a vector with the same direction as ${\displaystyle \overrightarrow{PQ}}$ but with length 1. Find a vector with the same direction as ${\displaystyle \overrightarrow{PQ}}$ but with length 4.

If $A,B$⁠, and $C$ are three points, find ${\displaystyle \overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}}$⁠.

Consider the 12 vectors that have their tails at the center of a clock and their respective heads at each of the 12 digits. What is the sum of these vectors? What if we remove the vector corresponding to 4 o'clock? What if, instead, all vectors have their tails at 12 o'clock, and their heads on the remaining digits?

Find $⟨\mathrm{1,1,1}⟩\cdot ⟨2,-\mathrm{3,4}⟩$⁠.

Find $⟨\mathrm{1,2,0}⟩\cdot ⟨\mathrm{0,0,57}⟩$⁠.

Find $⟨\mathrm{3,2,1}⟩\cdot ⟨\mathrm{0,1,0}⟩$⁠.

Find $⟨-1,-\mathrm{2,5}⟩\cdot ⟨\mathrm{1,0},-1⟩$⁠.

Find $⟨\mathrm{3,4,6}⟩\cdot ⟨\mathrm{2,3,4}⟩$⁠.

Find the cosine of the angle between $⟨\mathrm{1,2,3}⟩$ and $⟨\mathrm{1,1,1}⟩$⁠; use a calculator if necessary to find the angle.

Find the cosine of the angle between $⟨-1,-2,-3⟩$ and $⟨\mathrm{5,0,2}⟩$⁠; use a calculator if necessary to find the angle.

Find the cosine of the angle between $⟨\mathrm{47,100,0}⟩$ and $⟨\mathrm{0,0,5}⟩$⁠; use a calculator if necessary to find the angle.

Find the cosine of the angle between $⟨\mathrm{1,0,1}⟩$ and $⟨\mathrm{0,1,1}⟩$⁠; use a calculator if necessary to find the angle.

Find the cosine of the angle between $⟨\mathrm{2,0,0}⟩$ and $⟨-\mathrm{1,1},-1⟩$⁠; use a calculator if necessary to find the angle.

Find the angle between the diagonal of a cube and one of the edges adjacent to the diagonal.

Find the scalar and vector projections of $⟨\mathrm{1,2,3}⟩$ onto $⟨\mathrm{1,2,0}⟩$⁠.

Find the scalar and vector projections of $⟨\mathrm{1,1,1}⟩$ onto $⟨\mathrm{3,2,1}⟩$⁠.

A 20 pound object sits on a ramp at an angle of ${\displaystyle {30}^{\circ }}$ from the horizontal, as in figure 12.3.5. Find the force pulling the object down the ramp and the force pulling the object directly into the ramp.

A 20 pound object sits on a ramp at an angle of ${\displaystyle {45}^{\circ }}$ from the horizontal. Find the force pulling the object down the ramp and the force pulling the object directly into the ramp.

A force of 10 pounds is applied to a wagon, directed at an angle of ${\displaystyle {30}^{\circ }}$ from the horizontal, as shown. Find the component of this force pulling the wagon straight up, and the component pulling it horizontally along the ground.

A force of 15 pounds is applied to a wagon, directed at an angle of ${\displaystyle {45}^{\circ }}$ from the horizontal. Find the component of this force pulling the wagon straight up, and the component pulling it horizontally along the ground.

A force $𝐅$ is to be applied to a wagon, directed at an angle of ${\displaystyle {30}^{\circ }}$ from the horizontal. The resulting force pulling the wagon horizontally along the ground is to be 10 pounds. What is the magnitude of the required force $𝐅$⁠?

Use the dot product to find a non-zero vector $𝐰$ perpendicular to both $𝐮=⟨\mathrm{1,2},-3⟩$ and $𝐯=⟨\mathrm{2,0,1}⟩$⁠.

Let $𝐱=⟨\mathrm{1,1,0}⟩$ and $𝐲=⟨\mathrm{2,4,2}⟩$⁠. Find a unit vector that is perpendicular to both $𝐱$ and $𝐲$⁠.

Do the three points $(\mathrm{1,2,0})$⁠, $(-\mathrm{2,1,1})$⁠, and $(\mathrm{0,3},-1)$ form a right triangle?

Do the three points $(\mathrm{1,1,1})$⁠, $(\mathrm{2,3,2})$⁠, and $(\mathrm{5,0},-1)$ form a right triangle?

Find the cross product of $⟨\mathrm{1,1,1}⟩$ and $⟨\mathrm{1,2,3}⟩$⁠.

Find the cross product of $⟨\mathrm{1,0,2}⟩$ and $⟨-1,-\mathrm{2,4}⟩$⁠.

Find the cross product of $⟨-\mathrm{2,1,3}⟩$ and $⟨\mathrm{5,2},-1⟩$⁠.

Find the cross product of $⟨\mathrm{1,0,0}⟩$ and $⟨\mathrm{0,0,1}⟩$⁠.

Two vectors $𝐮$ and $𝐯$ are separated by an angle of $\pi /6$⁠, and $|𝐮|=2$ and $|𝐯|=3$⁠. Find $|𝐮\times 𝐯|$⁠.

Two vectors $𝐮$ and $𝐯$ are separated by an angle of $\pi /4$⁠, and $|𝐮|=3$ and $|𝐯|=7$⁠. Find $|𝐮\times 𝐯|$⁠.

Find the area of the parallelogram with vertices $(\mathrm{0,0})$⁠, $(\mathrm{1,2})$⁠, $(\mathrm{3,7})$⁠, and $(\mathrm{2,5})$⁠.

Find the area of the parallelogram with vertices $(0,-1)$⁠, $(\mathrm{3,4})$⁠, $(\mathrm{1,6})$⁠, and $(-\mathrm{2,1})$⁠.

Find the area of the triangle with vertices $(\mathrm{2,0,0})$⁠, $(\mathrm{1,3,4})$⁠, and $(-2,-\mathrm{1,1})$⁠.

Find the area of the triangle with vertices $(2,-\mathrm{2,1})$⁠, $(-\mathrm{3,2,3})$⁠, and $(\mathrm{3,3},-2)$⁠.

Find an equation of the plane containing $(\mathrm{6,2,1})$ and perpendicular to $⟨\mathrm{1,1,1}⟩$⁠.

Find an equation of the plane containing $(-\mathrm{1,2},-3)$ and perpendicular to $⟨\mathrm{4,5},-1⟩$⁠.

Find an equation of the plane containing $(\mathrm{1,2},-3)$⁠, $(\mathrm{0,1},-2)$ and $(\mathrm{1,2},-2)$⁠.

Find an equation of the plane containing $(\mathrm{1,0,0})$⁠, $(\mathrm{4,2,0})$ and $(\mathrm{3,2,1})$⁠.

Find an equation of the plane containing $(\mathrm{1,0,0})$ and the line $⟨\mathrm{1,0,2}⟩+t⟨\mathrm{3,2,1}⟩$⁠.

Find an equation of the plane containing the line of intersection of $x+y+z=1$ and $x-y+2z=2$⁠, and perpendicular to the plane $2x+3y-z=4$⁠.

Find an equation of the plane containing the line of intersection of $x+2y-z=3$ and $3x-y+4z=7$⁠, and perpendicular to the plane $6x-y+3z=16$⁠.

Find an equation of the plane containing the line of intersection of $x+3y-z=6$ and $2x+2y-3z=8$⁠, and perpendicular to the plane $3x+y-z=11$⁠.

Find an equation of the line through $(\mathrm{1,0,3})$ and $(\mathrm{1,2,4})$⁠.

Find an equation of the line through $(\mathrm{1,0,3})$ and perpendicular to the plane $x+2y-z=1$⁠.

Find an equation of the line through the origin and perpendicular to the plane $x+y-z=2$⁠.

Find $a$ and $c$ so that $(a,1,c)$ is on the line through $(\mathrm{0,2,3})$ and $(\mathrm{2,7,5})$⁠.

Determine whether the lines $⟨\mathrm{1,3},-1⟩+t⟨\mathrm{1,1,0}⟩$ and $⟨\mathrm{0,0,0}⟩+t⟨\mathrm{1,4,5}⟩$ are parallel, intersect, or neither.

Determine whether the lines $⟨\mathrm{1,0,2}⟩+t⟨-1,-\mathrm{1,2}⟩$ and $⟨\mathrm{4,4,2}⟩+t⟨\mathrm{2,2},-4⟩$ are parallel, intersect, or neither.

Determine whether the lines $⟨\mathrm{1,2},-1⟩+t⟨\mathrm{1,2,3}⟩$ and $⟨\mathrm{1,0,1}⟩+t⟨2/\mathrm{3,2,4}/3⟩$ are parallel, intersect, or neither.

Determine whether the lines $⟨\mathrm{1,1,2}⟩+t⟨\mathrm{1,2},-3⟩$ and $⟨\mathrm{2,3},-1⟩+t⟨\mathrm{2,4},-6⟩$ are parallel, intersect, or neither.

Find the distance from $(\mathrm{2,2,2})$ to $x+y+z=-1$⁠.

Find the distance from $(2,-1,-1)$ to $2x-3y+z=2$⁠.

Find the distance from $(2,-\mathrm{1,1})$ to $⟨\mathrm{2,2,0}⟩+t⟨\mathrm{1,2,3}⟩$⁠.

Find the distance from $(\mathrm{1,0,1})$ to $⟨\mathrm{3,2,1}⟩+t⟨2,-1,-2⟩$⁠.

Find the distance between the lines $⟨\mathrm{5,3,1}⟩+t⟨\mathrm{2,4,3}⟩$ and $⟨\mathrm{6,1,0}⟩+t⟨\mathrm{3,5,7}⟩$⁠.

Find the distance between the lines $⟨\mathrm{2,1,3}⟩+t⟨-\mathrm{1,2},-3⟩$ and $⟨1,-\mathrm{3,4}⟩+t⟨4,-\mathrm{4,1}⟩$⁠.

Find the distance between the lines $⟨\mathrm{1,2,3}⟩+t⟨2,-\mathrm{1,3}⟩$ and $⟨\mathrm{4,5,6}⟩+t⟨-\mathrm{4,2},-6⟩$⁠.

Find the distance between the lines $⟨\mathrm{3,2,1}⟩+t⟨\mathrm{1,4},-1⟩$ and $⟨\mathrm{3,1,3}⟩+t⟨\mathrm{2,8},-2⟩$⁠.

Find the cosine of the angle between the planes $x+y+z=2$ and $x+2y+3z=8$⁠.

Find the cosine of the angle between the planes $x-y+2z=2$ and $3x-2y+z=5$⁠.

Convert the following points in rectangular coordinates to cylindrical and spherical coordinates:

a. $(\mathrm{1,1,1})$

b. $(7,-\mathrm{7,5})$

c. $(\cos (1),\sin (1),1)$

d. $(\mathrm{0,0},-\pi )$

Find an equation for the sphere ${\displaystyle x^2+y^2+z^2=4}$ in cylindrical coordinates.

Find an equation for the $y$⁠-⁠$z$ plane in cylindrical coordinates.

Find an equation equivalent to ${\displaystyle x^2+y^2+2z^2+2z-5=0}$ in cylindrical coordinates.

Suppose the curve ${\displaystyle {\displaystyle z=e^{-x^2}}}$ in the $x$⁠-⁠$z$ plane is rotated around the $z$ axis. Find an equation for the resulting surface in cylindrical coordinates.

Suppose the curve ${\displaystyle z=x}$ in the $x$⁠-⁠$z$ plane is rotated around the $z$ axis. Find an equation for the resulting surface in cylindrical coordinates.

Find an equation for the plane $y=0$ in spherical coordinates.

Find an equation for the plane $z=1$ in spherical coordinates.

Find an equation for the sphere with radius 1 and center at $(\mathrm{0,1,0})$ in spherical coordinates.

Find an equation for the cylinder ${\displaystyle x^2+y^2=9}$ in spherical coordinates.

Suppose the curve ${\displaystyle z=x}$ in the $x$⁠-⁠$z$ plane is rotated around the $z$ axis. Find an equation for the resulting surface in spherical coordinates.

Extend exercises 6 and 11 by rotating the curve $z=mx$ around the $z$ axis and converting to both cylindrical and spherical coordinates.

Convert the spherical formula $\rho =\sin \theta \sin \varphi$ to rectangular coordinates and describe the surface defined by the formula (Hint: Multiply both sides by $\rho$⁠.)

We can describe points in the first octant by $x>0$⁠, $y>0$ and $z>0$⁠. Give similar inequalities for the first octant in cylindrical and spherical coordinates.