Chapter 13: Vector FunctionsContent check — Calculus (Guichard), Chapter 13: Vector Functions

Find a vector function for the curve of intersection of $x^2+y^2=9$ and $y+z=2$⁠.

A bug is crawling outward along the spoke of a wheel that lies along a radius of the wheel. The bug is crawling at 1 unit per second and the wheel is rotating at 1 radian per second. Suppose the wheel lies in the $y$⁠-⁠$z$ plane with center at the origin, and at time $t=0$ the spoke lies along the positive $y$ axis and the bug is at the origin. Find a vector function $𝐫(t)$ for the position of the bug at time $t$⁠.

Find ${𝐫}^{\prime }$ and $𝐓$ for $𝐫=⟨t^2,1,t⟩$⁠.

Find ${𝐫}^{\prime }$ and $𝐓$ for $𝐫=⟨\cos t,\sin 2t,t^2⟩$⁠.

Find ${𝐫}^{\prime }$ and $𝐓$ for $𝐫=$ $⟨\cos (e^t),$ $\sin (e^t),$ $\sin t⟩$⁠.

Find a vector function for the line tangent to the helix $⟨\cos t,\sin t,t⟩$ when $t=\pi /4$⁠.

Find a vector function for the line tangent to $⟨\cos t,\sin t,\cos 4t⟩$ when $t=\pi /3$⁠.

Find the cosine of the angle between the curves $⟨0,t^2,t⟩$ and $⟨\cos (\pi t/2),$ $\sin (\pi t/2),$ $t⟩$ where they intersect.

Find the cosine of the angle between the curves $⟨\cos t,$ $-$ $\sin (t)/4,$ $\sin t⟩$ and $⟨\cos t,\sin t,\sin (2t)⟩$ where they intersect.

A bug is crawling along the spoke of a wheel that lies along a radius of the wheel. The bug is crawling at 1 unit per second and the wheel is rotating at 1 radian per second. Suppose the wheel lies in the $y$⁠-⁠$z$ plane with center at the origin, and at time $t=0$ the spoke lies along the positive $y$ axis and the bug is at the origin. Find a vector function $𝐫(t)$ for the position of the bug at time $t$⁠, the velocity vector ${𝐫}^{\prime }(t)$⁠, the unit tangent $𝐓(t)$⁠, and the speed of the bug $|{𝐫}^{\prime }(t)|$⁠.

An object moves with velocity vector $⟨\cos t,\sin t,t⟩$⁠, starting at $(\mathrm{0,0,0})$ when $t=0$⁠. Find the function $𝐫$ giving its location.

An object moves with velocity vector $⟨t,t^2,-t⟩$⁠, starting at $(\mathrm{1,2,3})$ when $t=0$⁠. Find the function $𝐫$ giving its location.

An object moves with velocity vector $⟨t^2,\sin t,\cos t⟩$⁠, starting at $(-\mathrm{1,1,2})$ when $t=1$⁠. Find the function $𝐫$ giving its location.

The position function of a particle is given by $𝐫(t)=⟨t^2,5t,t^2-16t⟩$⁠, $t\ge 0$⁠. When is the speed of the particle a minimum?

A particle moves so that its position is given by $⟨\cos t,\sin t,\cos (6t)⟩$⁠. Find the maximum and minimum speeds of the particle.

An object moves with velocity vector $⟨t,t^2,\cos t⟩$⁠, starting at $(\mathrm{0,0,0})$ when $t=0$⁠. Find the function $𝐫$ giving its location.

Determine the point at which $𝐟(t)=⟨t,t^2,t^3⟩$ and $𝐠(t)=$ $⟨\cos (t),$ $\cos (2t),$ $t+$ $1⟩$ intersect, and find the angle between the curves at that point. (Hint: You'll need to set this one up like a line intersection problem, writing one in $s$ and one in $t$⁠.) If these two functions were the trajectories of two airplanes on the same scale of time, would the planes collide at their point of intersection? Explain.

Find the equation of the plane perpendicular to the curve $𝐫(t)=$ $⟨2\sin (3t),$ $t,$ $2\cos (3t)⟩$ at the point $(0,\pi ,-2)$⁠.

Find the equation of the plane perpendicular to $⟨\cos t,\sin t,\cos (6t)⟩$ when $t=\pi /4$⁠.

At what point on the curve $𝐫(t)=⟨t^3,3t,t^4⟩$ is the plane perpendicular to the curve also parallel to the plane $6x+6y-8z=1$⁠?

Find the equation of the line tangent to $⟨\cos t,\sin t,\cos (6t)⟩$ when $t=\pi /4$⁠.

Find the length of $⟨3\cos t,2t,3\sin t⟩$⁠, $t\in [\mathrm{0,2}\pi ]$⁠.

Find the length of $⟨t^2,2,t^3⟩$⁠, $t\in [\mathrm{0,1}]$⁠.

Find the length of $⟨t^2,\sin t,\cos t⟩$⁠, $t\in [\mathrm{0,1}]$⁠.

Find the length of the curve $y=x^{3/2}$⁠, $x\in [\mathrm{1,9}]$⁠.

Set up an integral to compute the length of $⟨\cos t,\sin t,e^t⟩$⁠, $t\in [\mathrm{0,5}]$⁠. (It is tedious but not too difficult to compute this integral.)

Find the curvature of $⟨t,t^2,t⟩$⁠.

Find the curvature of $⟨t,t^2,t^2⟩$⁠.

Find the curvature of $⟨t,t^2,t^3⟩$⁠.

Find the curvature of $y=x^4$ at $(\mathrm{1,1})$⁠.

Let $𝐫=⟨\cos t,\sin t,t⟩$⁠. Compute $𝐯$⁠, $𝐚$⁠, $a_T$⁠, and $a_N$⁠.

Let $𝐫=⟨\cos t,\sin t,t^2⟩$⁠. Compute $𝐯$⁠, $𝐚$⁠, $a_T$⁠, and $a_N$⁠.

Let $𝐫=⟨\cos t,\sin t,e^t⟩$⁠. Compute $𝐯$⁠, $𝐚$⁠, $a_T$⁠, and $a_N$⁠.

Let $𝐫=⟨t^2,2t-\mathrm{3,3}{t}^2-3t⟩$⁠. Compute $𝐯$⁠, $𝐚$⁠, $a_T$⁠, and $a_N$⁠.

Let $𝐫=⟨e^t,\sin t,e^t⟩$⁠. Compute $𝐯$⁠, $𝐚$⁠, $a_T$⁠, and $a_N$⁠.

Suppose an object moves so that its acceleration is given by $𝐚=⟨-3\cos t,-2\sin t,0⟩$⁠. At time $t=0$ the object is at $(\mathrm{3,0,0})$ and its velocity vector is $⟨\mathrm{0,2,0}⟩$⁠. Find $𝐯(t)$ and $𝐫(t)$ for the object.

Suppose an object moves so that its acceleration is given by $𝐚=⟨-3\cos t,-2\sin t,0⟩$⁠. At time $t=0$ the object is at $(\mathrm{3,0,0})$ and its velocity vector is $⟨\mathrm{0,2.1,0}⟩$⁠. Find $𝐯(t)$ and $𝐫(t)$ for the object.

Suppose an object moves so that its acceleration is given by $𝐚=⟨-3\cos t,-2\sin t,0⟩$⁠. At time $t=0$ the object is at $(\mathrm{3,0,0})$ and its velocity vector is $⟨\mathrm{0,2,1}⟩$⁠. Find $𝐯(t)$ and $𝐫(t)$ for the object.

Suppose an object moves so that its acceleration is given by $𝐚=⟨-3\cos t,-2\sin t,0⟩$⁠. At time $t=0$ the object is at $(\mathrm{3,0,0})$ and its velocity vector is $⟨\mathrm{0,2.1,1}⟩$⁠. Find $𝐯(t)$ and $𝐫(t)$ for the object.