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

$\kappa = \frac{\|\vec{r}'(t) \times \vec{r}''(t)\|}{\|\vec{r}'(t)\|^3}$

Question 1

Given the vector function $\vec{r}(t) = (t-3)\vec{i} + t\vec{j} + t^2\vec{k}$, what is $\vec{r}'(t)$?

Accepted Answer

$\vec{i} + \vec{j} + 2t\vec{k}$

Question 2

Find the equation of the tangent line to the curve $\vec{r}(t) = (t-3)\vec{i} + t\vec{j} + t^2\vec{k}$ at $t=0$.

Accepted Answer

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

Question 3

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

Accepted Answer

$(-4, 2, \frac{8}{3})$

Question 4

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

Accepted Answer

$1$

Question 5

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

Accepted Answer

$(2, 1.5, 0.5)$

Question 6

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

Accepted Answer

$(0.5, 1.1667, 2.875)$

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

$3\sqrt{37}$

Question 8

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

Accepted Answer

$\frac{1}{\sqrt{37}}(-6\sin(6t), 6\cos(6t), 1)$

Question 9

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

Accepted Answer

$(-\cos(6t), -\sin(6t), 0)$

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

$0$

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

$25$

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 a curve with constant speed. Which of the following is true about its acceleration vector?

Accepted Answer

The acceleration is always perpendicular to the velocity vector.

Question 14

If $\vec{r}(t)$ is a position vector of a particle, which of the following gives the velocity vector?

Accepted Answer

$\vec{v}(t) = \frac{d\vec{r}}{dt}$

Question 15

For a curve $\vec{r}(t)$, which formula gives the unit tangent vector $\vec{T}(t)$?

Accepted Answer

$\vec{T}(t) = \frac{\vec{r}'(t)}{\|\vec{r}'(t)\|}$

Question 16

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

Accepted Answer

$\sqrt{2}$

Question 17

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

Accepted Answer

$\kappa = \frac{\|\vec{r}'(t) 	imes \vec{r}''(t)\|}{\|\vec{r}'(t)\|^3}$

Question 18

If a particle's velocity vector is always tangent to its path, which of the following is true?

Accepted Answer

The velocity vector is always parallel to the unit tangent vector.

Question 19

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

Accepted Answer

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

Vector Functions, Tangent Lines, Curvature, and Motion in Space

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