Q: Given $$\vec{A} = 3\hat{i} + 4\hat{j}$$ and $$\vec{B} = 5\hat{i} + 12\hat{j}$$, what is $$\vec{A} \cdot \vec{B}$$?

$$3 \times 5 + 4 \times 12 = 15 + 48 = 63$$

Q: Which of the following statements about the cross product $$\vec{A} \times \vec{B} is $$correct?

The direction of $$\vec{A} \times \vec{B} is $$given by the right-hand rule.

Q: If $$\vec{A}$$ and $$\vec{B}$$ are parallel, what is $$\vec{A} \cdot \vec{B}$$ and $$\vec{A} \times \vec{B}$$?

$$\vec{A} \cdot \vec{B} is $$maximum, $$\vec{A} \times \vec{B} = 0$$

Q: Given $$\vec{A} = 2\hat{i} + 3\hat{j} + 4\hat{k}$$ and $$\vec{B} = \hat{i} - 2\hat{j} + 2\hat{k}$$, what is the $$k-$$component of $$\vec{A} \times \vec{B}$$?

$$A_x B_y - A_y B_x = 2 \times (-2) - 3 \times 1 = -4 - 3 = -7$$

Q: A particle moves along the x-axis with position $$x(t) = 4t^2$ - 2t + 1. $$What is its instantaneous velocity at $$t = 3\,\mathrm{s}$$?

$$v(t) = \frac{dx}{dt} = 8t - 2;\; v(3) = 8 \times 3 - 2 = 22\,\mathrm{m/s}$$

Question 1

A vector $$\vec{A}$$ has a magnitude of $$10\,\mathrm{m}$$ and makes an angle of $$30^\circ$$ with the positive x-axis. What are the x and y components of $$\vec{A}$$?

Accepted Answer

$$A_x = 10 \cos 30^\circ = 8.66\,\mathrm{m},\; A_y = 10 \sin 30^\circ = 5.0\,\mathrm{m}$$

Question 2

Given $$\vec{A} = 3\hat{i} + 4\hat{j}$$ and $$\vec{B} = 5\hat{i} + 12\hat{j}$$, what is $$\vec{A} \cdot \vec{B}$$?

Accepted Answer

$$3 	imes 5 + 4 	imes 12 = 15 + 48 = 63$$

Question 3

Which of the following statements about the cross product $$\vec{A} 	imes \vec{B} is $$correct?

Accepted Answer

The direction of $$\vec{A} 	imes \vec{B} is $$given by the right-hand rule.

Question 4

If $$\vec{A}$$ and $$\vec{B}$$ are parallel, what is $$\vec{A} \cdot \vec{B}$$ and $$\vec{A} 	imes \vec{B}$$?

Accepted Answer

$$\vec{A} \cdot \vec{B} is $$maximum, $$\vec{A} 	imes \vec{B} = 0$$

Question 5

A car moves from $$x_1 = 2$$\,\mathrm{m}$$ at t_1$ = 0\,\mathrm{s} to$$ $$x_2 = 10$$\,\mathrm{m}$$ at t_2$ = 4\,\mathrm{s}. $$What is its average velocity?

Accepted Answer

$$\frac{10 - 2}{4 - 0} = 2\,\mathrm{m/s}$$

Question 6

Which of the following is true about the dot product $$\vec{A} \cdot \vec{B}$$?

Accepted Answer

It is maximum when $$\vec{A}$$ and $$\vec{B}$$ are parallel.

Question 7

Given $$\vec{A} = 2\hat{i} + 3\hat{j} + 4\hat{k}$$ and $$\vec{B} = \hat{i} - 2\hat{j} + 2\hat{k}$$, what is the $$k-$$component of $$\vec{A} 	imes \vec{B}$$?

Accepted Answer

$$A_x B_y - A_y B_x = 2 	imes (-2) - 3 	imes 1 = -4 - 3 = -7$$

Question 8

A runner travels 1 km east, then 1 km north, then 1 km west. What is the magnitude of the displacement from the starting point?

Accepted Answer

$$1\,\mathrm{km}$$ north

Question 9

Which of the following best describes the difference between average speed and average velocity?

Accepted Answer

Average speed is the total distance divided by total time; average velocity is the displacement divided by total time.

Question 10

A particle moves along the x-axis with position $$x(t) = 4t^2$ - 2t + 1. $$What is its instantaneous velocity at $$t = 3\,\mathrm{s}$$?

Accepted Answer

$$v(t) = \frac{dx}{dt} = 8t - 2;\; v(3) = 8 	imes 3 - 2 = 22\,\mathrm{m/s}$$

Question 11

If $$\vec{A} = 2\hat{i} + 3\hat{j}$$ and $$\vec{B} = -\hat{i} + 4\hat{j}$$, what is the angle $$\phi$$ between them?

Accepted Answer

$$\cos \phi = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}||\vec{B}|}$$, so $$\phi = \cos$$^{-1}$$\left(\frac{2 	imes (-1) + 3 	imes 4}{\sqrt{2^2$ + 3^2$} \sqrt{(-1)^2$ + 4^2$}}ight) = \cos$$^{-1}$$\left(\frac{-2 + 12}{\sqrt{13} \sqrt{17}}ight) = \cos$$^{-1}$$\left(\frac{10}{\sqrt{221}}ight)$$

Question 12

A car's velocity changes from $$10\,\mathrm{m/s} to$$ $$-5\,\mathrm{m/s} in$$ $$3\,\mathrm{s}. $$What is its average acceleration?

Accepted Answer

$$a_{av} = \frac{-5 - 10}{3} = \frac{-15}{3} = -5\,\mathrm{m/s^2$}$$

Question 13

Which of the following is true about the commutativity of vector products?

Accepted Answer

$$\vec{A} 	imes \vec{B} = -\vec{B} 	imes \vec{A}$$

Question 14

A position vs. time graph shows a straight line with positive slope. What does this indicate about the object's motion?

Accepted Answer

The object moves with constant positive velocity.

Question 15

If a particle's velocity and acceleration are in opposite directions, what happens to its speed?

Accepted Answer

The particle slows down.

Question 16

Given $$\vec{A} = 2\hat{i} + 3\hat{j}$$ and $$\vec{B} = 4\hat{i} - \hat{j}$$, what is $$\vec{A} 	imes \vec{B}$$?

Accepted Answer

$$(2)(-1) - (3)(4) = -2 - 12 = -14\hat{k}$$

Question 17

A ball is thrown straight up with an initial velocity of $$20\,\mathrm{m/s}. $$What is its velocity after $$2\,\mathrm{s}$$? (Assume $$g = 9.8\,\mathrm{m/s^2$}$$ downward.)

Accepted Answer

$$v = v_0$ - gt = 20 - 9.8 	imes 2 = 0.4\,\mathrm{m/s}$$

Question 18

Which of the following is a correct interpretation of the dot product $$\vec{A} \cdot \vec{B}$$?

Accepted Answer

It equals the magnitude of $$\vec{A}$$ times the component of $$\vec{B} in $$the direction of $$\vec{A}.$$

Vector Operations and 1D Kinematics: Study Notes for College Physics

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