Question 1

Given the vector function $$\vec{r}(t) = (t-3)\hat{i} + t\hat{j} + t^2$\hat{k}$$, what is the velocity vector $$\vec{v}(t)$$?

Accepted Answer

$$\vec{v}(t) = \hat{i} + \hat{j} + 2t\hat{k}$$

Question 2

For the vector function $$\vec{r}(t) = (t-3)\hat{i} + t\hat{j} + t^2$\hat{k}$$, what is the equation of the tangent line at $$t=0$$?

Accepted Answer

$$\vec{l}(s) = (-3, 0, 0) + s(1, 1, 0)$$

Question 3

Evaluate $$\$$int_0$$^3 \vec{r}(t)\,dt$$ for $$\vec{r}(t) = (t-3)\hat{i} + t\hat{j} + t^2$\hat{k}.$$

Accepted Answer

$$(-4.5, 4.5, 9) $$

Question 4

Given $$\vec{r}(t) = (t-3)\hat{i} + t\hat{j} + t^2$\hat{k}$$, what is the curvature $$\kappa(0) at $$the point $$(-3, 0, 0)$$?

Accepted Answer

$$\kappa(0) = \dfrac{2}{(1^2$ + 1^2$ + 0^2$)^{3/2}$$} = \dfrac{1}{2}$

Question 5

A particle has acceleration $$\vec{a}(t) = (1, t^2$, t)$$, position $$\vec{r}(1) = (1, 2, 3)$$, and velocity $$\vec{v}(1) = (3, 2, 1). $$What is the original velocity vector $$\vec{v}(0)$$?

Accepted Answer

$$(2, 1.5, 0.5)$$

Question 6

A particle has acceleration $$\vec{a}(t) = (1, t^2$, t)$$, position $$\vec{r}(1) = (1, 2, 3)$$, and velocity $$\vec{v}(1) = (3, 2, 1). $$What is the original position vector $$\vec{r}(0)$$?

Accepted Answer

$$(0, 1.5, 2.5)$$

Question 7

Given $$\vec{r}(t) = (\cos(6t), \sin(6t), t-6)$$, what is the arc length from $$t=0 to$$ $$t=3$$?

Accepted Answer

$$\$$int_0$$^3 \sqrt{36 + 1}\,dt = 3\sqrt{37}$$

Question 8

For $$\vec{r}(t) = (\cos(6t), \sin(6t), t-6)$$, what is the unit tangent vector $$\vec{T}(t)$$?

Accepted Answer

$$\vec{T}(t) = \dfrac{(-6\sin(6t), 6\cos(6t), 1)}{\sqrt{36 + 1}}$$

Question 9

For $$\vec{r}(t) = (\cos(6t), \sin(6t), t-6)$$, what is the principal unit normal vector $$\vec{N}(t)$$?

Accepted Answer

$$\vec{N}(t) = (-\cos(6t), -\sin(6t), 0)$$

Question 10

Given $$\vec{r}(t) = (\cos(5t), \sin(5t), t)$$, what is the tangential component of acceleration $$a_T at $$any $$t$$?

Accepted Answer

$$a_T = 0$$

Question 11

Given $$\vec{r}(t) = (\cos(5t), \sin(5t), t)$$, what is the normal component of acceleration $$a_N at $$any $$t$$?

Accepted Answer

$$a_N = 5$$

Question 12

Which of the following best describes the geometric meaning of the curvature $$\kappa of a $$space curve at a point?

Accepted Answer

It measures how quickly the curve is changing direction at that point.

Question 13

Suppose a particle moves along $$\vec{r}(t) = (t, t^2$, t^3$). $$What is the magnitude of the velocity at $$t=1$$?

Accepted Answer

$$\sqrt{1^2$ + (2 \cdot 1)^2$ + (3 \cdot 1^2$)^2$$} = \sqrt{1 + 4 + 9} = \sqrt{14}$

Question 14

A car travels along a path given by $$\vec{r}(t) = (t, \sin t, 0). $$What is the speed of the car at $$t=\pi$$?

Accepted Answer

$$\sqrt{1 + \cos^2$ \pi} = \sqrt{2}$$

Question 15

For a vector function $$\vec{r}(t)$$, which of the following is the correct formula for the unit tangent vector $$\vec{T}(t)$$?

Accepted Answer

$$\vec{T}(t) = \dfrac{\vec{v}(t)}{\|\vec{v}(t)\|}$$

Question 16

If $$\vec{r}(t) is a $$position vector, which of the following gives the principal unit normal vector $$\vec{N}(t)$$?

Accepted Answer

$$\vec{N}(t) = \dfrac{d\vec{T}/dt}{\|d\vec{T}/dt\|}$$

Question 17

A drone flies along $$\vec{r}(t) = (2t, 3t^2$, 4t^3$). $$What is the magnitude of its acceleration at $$t=1$$?

Accepted Answer

$$\sqrt{0^2$ + 6^2$ + 24^2$} = \sqrt{36 + 576} = 24.74$$

Question 18

Which of the following is the correct formula for the curvature $$\kappa of a $$space curve $$\vec{r}(t)$$?

Accepted Answer

$$\kappa = \dfrac{\|\vec{v}(t) 	imes \vec{a}(t)\|}{\|\vec{v}(t)\|^3}$$

Question 19

A roller coaster moves along $$\vec{r}(t) = (t, t^2$, \sin t). $$What is the binormal vector $$\vec{B}(t) at$$ $$t=0$$?

Accepted Answer

$$\vec{B}(0) = (0, 0, 1)$$

Vector Functions, Curves, and Motion in Calculus III: Study Notes

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