Q: For vector functions $$\mathbf{f}(t)$$ and $$\mathbf{g}(t)$$, which formula correctly expresses the derivative of their cross product?

$$\frac{d}{dt}[\mathbf{f}(t) \times \mathbf{g}(t)] = \frac{d\mathbf{f}}{dt} \times \mathbf{g}(t) + \mathbf{f}(t) \times \frac{d\mathbf{g}}{dt}$$

Q: If $$\mathbf{v}(t) is $$the velocity vector of a particle, which formula gives the normal component of acceleration $$a_N(t)$$?

$$a_N(t) = \frac{\|\mathbf{v}(t) \times \mathbf{a}(t)\|}{\|\mathbf{v}(t)\|}$$

Q: If $$\mathbf{f}(t) is a $$vector function, what is $$\frac{d}{dt}[\mathbf{f}(t) \times \frac{d\mathbf{f}}{dt}]$$?

$$\mathbf{f}(t) \times \frac{d^2$\mathbf{f}}{dt^2$}$$

Question 1

Given two vector functions $$\mathbf{f}(t)$$ and $$\mathbf{g}(t)$$, which of the following correctly expresses the derivative of their dot product with respect to $$t$$?

Accepted Answer

$$\frac{d}{dt}[\mathbf{f}(t) \cdot \mathbf{g}(t)] = \frac{d\mathbf{f}}{dt} \cdot \mathbf{g}(t) + \mathbf{f}(t) \cdot \frac{d\mathbf{g}}{dt}$$

Question 2

If $$\mathbf{f}(t) = \langle t^2$, 1-t, \sqrt{t} \rangle$$, what is the magnitude $$\|\mathbf{f}(t)\|$$?

Accepted Answer

$$\sqrt{t^4$ + t^2$ - t + 1}$$

Question 3

Which statement is TRUE about the derivative of a unit vector function $$\hat{\mathbf{f}}(t)$$?

Accepted Answer

$$\hat{\mathbf{f}}(t) is $$always perpendicular to $$\frac{d\hat{\mathbf{f}}}{dt}.$$

Question 4

Given $$\mathbf{f}(t) = \langle \sin t, \cos t, 1 \rangle$$, what is the magnitude of its unit vector $$\hat{\mathbf{f}}(t)$$?

Accepted Answer

$$1$$

Question 5

Which formula gives the derivative of the magnitude of a vector function $$\mathbf{f}(t)$$?

Accepted Answer

$$\frac{d}{dt}\|\mathbf{f}(t)\| = \frac{1}{\|\mathbf{f}(t)\|} \left( \mathbf{f}(t) \cdot \frac{d\mathbf{f}}{dt} \right)$$

Question 6

For vector functions $$\mathbf{f}(t)$$ and $$\mathbf{g}(t)$$, which formula correctly expresses the derivative of their cross product?

Accepted Answer

$$\frac{d}{dt}[\mathbf{f}(t) 	imes \mathbf{g}(t)] = \frac{d\mathbf{f}}{dt} 	imes \mathbf{g}(t) + \mathbf{f}(t) 	imes \frac{d\mathbf{g}}{dt}$$

Question 7

A particle moves along a path given by $$\mathbf{r}(t) = \langle t, \frac{1}{2}t^2$, 0 \rangle. $$What is the equation of its trajectory in the $$xy-$$plane?

Accepted Answer

$$y = \frac{1}{2}x^2$

Question 8

Given $$\mathbf{v}(t) = \langle 1, t, 0 \rangle$$ and $$\mathbf{a}(t) = \langle 0, 1, 0 \rangle$$, what is the tangential component of acceleration $$a_T(t)$$?

Accepted Answer

$$a_T(t) = \frac{t}{\sqrt{t^2$+1}}$$

Question 9

For the same particle as above, what is the normal component of acceleration $$a_N(t)$$?

Accepted Answer

$$a_N(t) = \frac{1}{\sqrt{t^2$+1}}$$

Question 10

If $$\mathbf{f}(t) = \langle 2t, 3t^2$, t^3$ angle$$ and $$\mathbf{g}(t) = \langle 4t^2$, 2t, 1 angle$$, what is the $$y-$$component of $$\frac{d}{dt}[\mathbf{f}(t) 	imes \mathbf{g}(t)]$$?

Accepted Answer

$$20t^4 - 2$

Question 11

Which of the following statements about the cross product of a unit vector function and its normalized derivative is correct?

Accepted Answer

The magnitude is always one.

Question 12

A particle moves along a parabolic path in the $$xy-$$plane. At $$t=1 s$$, its acceleration vector is $$\mathbf{a}(1) = \langle 0, 1, 0 \rangle. $$What are the tangential and normal unit vectors at this instant?

Accepted Answer

$$\hat{T}(1) = \langle 1/\sqrt{2}, 1/\sqrt{2}, 0 \rangle$$, $$\hat{N}(1) = \langle -1/\sqrt{2}, 1/\sqrt{2}, 0 \rangle$$

Question 13

If $$\mathbf{v}(t) is $$the velocity vector of a particle, which formula gives the normal component of acceleration $$a_N(t)$$?

Accepted Answer

$$a_N(t) = \frac{\|\mathbf{v}(t) 	imes \mathbf{a}(t)\|}{\|\mathbf{v}(t)\|}$$

Question 14

Which of the following is a property of the dot product used in differentiating the magnitude of a vector function?

Accepted Answer

The dot product is commutative.

Question 15

If $$\mathbf{f}(t) is a $$vector function, what is $$\frac{d}{dt}[\mathbf{f}(t) 	imes \frac{d\mathbf{f}}{dt}]$$?

Accepted Answer

$$\mathbf{f}(t) 	imes \frac{d^2$\mathbf{f}}{dt^2$}$$

Question 16

A particle's velocity vector is $$\mathbf{v}(t)$$ and its acceleration is $$\mathbf{a}(t). $$Which formula gives the tangential component of acceleration?

Accepted Answer

$$a_T(t) = \frac{\mathbf{v}(t) \cdot \mathbf{a}(t)}{\|\mathbf{v}(t)\|}$$

Question 17

If $$\mathbf{f}(t) is a $$vector function, which of the following is TRUE about $$\frac{d}{dt}\|\mathbf{f}(t)\|$$ when $$\mathbf{f}(t) = 0$$?

Accepted Answer

The derivative is undefined.

Question 18

Which of the following best describes the direction of the unit normal vector $$\hat{N} in $$the context of a particle's motion?

Accepted Answer

It points in the direction the particle's path is turning.

Differentiation of Dot and Cross Products of Vector Functions; Tangential and Normal Components of Acceleration

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