Q: A car travels 20 km north and then 35 km in a direction 60° west of north. What is the magnitude of the car’s resultant displacement?

$$\sqrt{20^2$ + 35^2$ - 2 \times 20 \times 35 \times \cos 120^\circ} = 48.2 km$$

Q: Given vectors $$\vec{A} = 2\hat{i} + 2\hat{j}$$ and $$\vec{B} = 2\hat{i} - 4\hat{j}$$, what is the direction of the resultant vector $$\vec{R} = \vec{A} + \vec{B}$$ with respect to the x-axis?

$$\theta = \arctg\left(\frac{-2}{4}\right) = -27^\circ$$

Q: What is the scalar product (dot product) of two vectors $$\vec{A}$$ and $$\vec{B}$$?

$$\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta$$

Q: Which of the following is the correct formula for the vector product (cross product) of two vectors $$\vec{A}$$ and $$\vec{B}$$?

$$\vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin \theta \, \hat{n}$$

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

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

Q: Which of the following is the correct expression for the components of a vector $$\vec{A} in $$terms of its magnitude $$A$$ and angle $$\theta$$ with the x-axis?

$$A_x = A \cos \theta$$, $$A_y = A \sin \theta$$

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

$$\theta = \cos$$^{-1}$$\left(\frac{2(-4) + 3(2) + 1(-1)}{\sqrt{2^2$ + 3^2$ + 1^2$} \sqrt{(-4)^2$ + 2^2$ + (-1)^2$}}\right)$$

Question 1

Which of the following statements about the addition of two vectors is correct?

Accepted Answer

The sum of two vectors is independent of the order of addition.

Question 2

A car travels 20 km north and then 35 km in a direction 60° west of north. What is the magnitude of the car’s resultant displacement?

Accepted Answer

$$\sqrt{20^2$ + 35^2$ - 2 	imes 20 	imes 35 	imes \cos 120^\circ} = 48.2 km$$

Question 3

If vector $$\vec{A}$$ has components $$A_x = 3$$ and $$A_y = 4$$, what is the magnitude of $$\vec{A}$$?

Accepted Answer

$$\sqrt{3^2$ + 4^2$} = 5$$

Question 4

Which of the following is true about unit vectors?

Accepted Answer

Unit vectors have a magnitude of exactly 1.

Question 5

Given vectors $$\vec{A} = 2\hat{i} + 2\hat{j}$$ and $$\vec{B} = 2\hat{i} - 4\hat{j}$$, what is the direction of the resultant vector $$\vec{R} = \vec{A} + \vec{B}$$ with respect to the x-axis?

Accepted Answer

$$	heta = \arctg\left(\frac{-2}{4}ight) = -27^\circ$$

Question 6

What is the scalar product (dot product) of two vectors $$\vec{A}$$ and $$\vec{B}$$?

Accepted Answer

$$\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos 	heta$$

Question 7

If $$\vec{A}$$ and $$\vec{B}$$ are perpendicular, what is their scalar product?

Accepted Answer

$$\vec{A} \cdot \vec{B} = 0$$

Question 8

Which of the following is the correct formula for the vector product (cross product) of two vectors $$\vec{A}$$ and $$\vec{B}$$?

Accepted Answer

$$\vec{A} 	imes \vec{B} = |\vec{A}| |\vec{B}| \sin 	heta \, \hat{n}$$

Question 9

If a vector $$\vec{A} is $$multiplied by a negative scalar $$-m$$, what happens to the direction and magnitude of the resulting vector?

Accepted Answer

The direction is reversed and the magnitude is $$m|\vec{A}|.$$

Question 10

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

Accepted Answer

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

Question 11

Which of the following is the correct expression for the components of a vector $$\vec{A} in $$terms of its magnitude $$A$$ and angle $$	heta$$ with the x-axis?

Accepted Answer

$$A_x = A \cos 	heta$$, $$A_y = A \sin 	heta$$

Question 12

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

Accepted Answer

$$	heta = \cos$$^{-1}$$\left(\frac{2(-4) + 3(2) + 1(-1)}{\sqrt{2^2$ + 3^2$ + 1^2$} \sqrt{(-4)^2$ + 2^2$ + (-1)^2$}}ight)$$

Question 13

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

Accepted Answer

It results in a vector perpendicular to both $$\vec{A}$$ and $$\vec{B}.$$

Question 14

If $$\vec{A}$$ and $$\vec{B}$$ are parallel, what is the value of their cross product $$\vec{A} 	imes \vec{B}$$?

Accepted Answer

$$\vec{A} 	imes \vec{B} = 0$$

Question 15

A person walks 4 km northwest (20° with north), then 5 km south, and finally 3 km east. What is the magnitude of the net displacement?

Accepted Answer

$$\sqrt{(-1.368)^2$ + (8.758)^2$} \approx 9 km$$

Question 16

Which of the following is the correct formula for the magnitude of the resultant vector $$\vec{R}$$ given its components $$R_x$$ and $$R_y$$?

Accepted Answer

$$|\vec{R}| = \sqrt{R_x^2$ + R_y^2$}$$

Question 17

If $$\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$$ and $$\vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k}$$, what is the scalar product $$\vec{A} \cdot \vec{B}$$?

Accepted Answer

$$A_x B_x + A_y B_y + A_z B_z$$

Question 18

Which of the following is true about the magnitude of a unit vector?

Accepted Answer

$$|\hat{i}| = |\hat{j}| = |\hat{k}| = 1$$

Fundamental Properties of Vectors: A Mini-Textbook Study Guide

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