Question 1

Which of the following physical quantities is a vector?

Accepted Answer

Velocity

Question 2

Which operation is used to find the resultant vector when two vectors are perpendicular?

Accepted Answer

Triangle math (Pythagorean theorem)

Question 3

A delivery truck travels 8 miles in the x-direction, 5 miles in the y-direction, and 4 miles again in the x-direction. What is the magnitude of its final displacement from the origin?

Accepted Answer

$$\sqrt{(8+4)^2$ + 5^2$} = \sqrt{144 + 25} = \sqrt{169} = 13$$ miles

Question 4

If $$\vec{A} = 2\hat{i} + 3\hat{j} + 4\hat{k}$$, what is the magnitude of $$\vec{A}$$?

Accepted Answer

$$\sqrt{2^2$ + 3^2$ + 4^2$} = \sqrt{4 + 9 + 16} = \sqrt{29} \approx 5.39$$

Question 5

When subtracting vectors graphically, what is the correct method?

Accepted Answer

Add the negative of the vector to be subtracted

Question 6

Given $$\vec{B} = 7\hat{i} - 3\hat{j}$$ and $$\vec{C} = -2\hat{i} + 5\hat{j}$$, what is $$\vec{B} - \vec{C}$$?

Accepted Answer

$$(7 - (-2))\hat{i} + (-3 - 5)\hat{j} = 9\hat{i} - 8\hat{j}$$

Question 7

If a vector $$\vec{A}$$ has a y-component of 12 and makes an angle of $$67.4^\circ$$ with the positive x-axis, what is the magnitude of $$\vec{A}$$?

Accepted Answer

$$\frac{12}{\sin(67.4^\circ)} \approx 13.0$$

Question 8

A vector $$\vec{F} is 65 $$units long, directed $$30.5^\circ$$ below the positive x-axis. What is the x-component $$F_x$$?

Accepted Answer

$$65 \cos(30.5^\circ) \approx 56.0$$

Question 9

Given $$A_x = -77 cm $$and $$A_y = 36 cm$$, what is the magnitude of vector $$\vec{A}$$?

Accepted Answer

$$\sqrt{(-77)^2$ + 36^2$} = \sqrt{5929 + 1296} = \sqrt{7225} = 85 cm$$

Question 10

A small helicopter travels 225 m across a city in a direction $$53.1^\circ$$ south of east. What is the y-component of its trip?

Accepted Answer

$$225 \sin(53.1^\circ) \approx 180 m$$

Question 11

Which of the following is the correct unit vector form for $$\vec{A} = 4\hat{i} + 2\hat{j}$$?

Accepted Answer

$$\frac{4}{\sqrt{4^2$ + 2^2$}}\hat{i} + \frac{2}{\sqrt{4^2$ + 2^2$}}\hat{j}$$

Question 12

What is the dot product $$\vec{A} \cdot \vec{B} if$$ $$\vec{A} = 6\hat{i} - 3\hat{j} - 4\hat{k}$$ and $$\vec{B} = (8.6\hat{i} + 2\hat{j} - 5\hat{k})$$?

Accepted Answer

$$6 	imes 8.6 + (-3) 	imes 2 + (-4) 	imes (-5) = 51.6 - 6 + 20 = 65.6$$

Question 13

If $$\vec{A} = 2\hat{i} + 3\hat{j}$$ and $$\vec{B} = 4\hat{i} + 5\hat{j}$$, what is the angle $$	heta$$ between $$\vec{A}$$ and $$\vec{B}$$?

Accepted Answer

$$\cos$$^{-1}$$\left(\frac{2 	imes 4 + 3 	imes 5}{\sqrt{2^2$ + 3^2$} \sqrt{4^2$ + 5^2$}}ight) = \cos$$^{-1}$$\left(\frac{8 + 15}{\sqrt{13} \sqrt{41}}ight) \approx 6.4^\circ$$

Question 14

What is the result of the cross product $$\vec{A} 	imes \vec{B}$$ for $$\vec{A} = 2\hat{i} + 3\hat{j}$$ and $$\vec{B} = 4\hat{i} + 5\hat{j}$$?

Accepted Answer

$$(2 	imes 5 - 3 	imes 4)\hat{k} = (10 - 12)\hat{k} = -2\hat{k}$$

Question 15

Two vectors $$\vec{A}$$ and $$\vec{B}$$ have a scalar product of $$-6$$ and a vector product magnitude of $$12. $$What is the angle between them?

Accepted Answer

$$	heta = \cos$$^{-1}$$\left(\frac{-6}{|A||B|}ight)$$ and $$\sin$$^{-1}$$\left(\frac{12}{|A||B|}ight)$$, so $$	heta = 120^\circ$$

Question 16

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

Accepted Answer

It produces a vector perpendicular to both $$\vec{A}$$ and $$\vec{B}$$

Vectors: Concepts, Operations, and Applications in Physics today

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