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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