Q: Calculate the dot product $$\vec{A} \cdot \vec{B}$$ for $$\vec{A} = 2\hat{i} + 3\hat{j}$$ and $$\vec{B} = \hat{i} + 2\hat{j}.$$

$$2 \times 1 + 3 \times 2 = 8$$

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

$$|4 \times 3 - 3 \times (-2)| = 18$$

Q: Two vectors $$\vec{A}$$ and $$\vec{B}$$ have a scalar product of $$-8$$ and a vector product of magnitude $$12. $$What is the angle between them?

$$\arccos\left(\frac{-8}{|A||B|}\right)$$ and $$\arcsin\left(\frac{12}{|A||B|}\right)$$, which are consistent for $$\theta = 123.7°$$

Question 1

Which of the following quantities is a vector?

Accepted Answer

Displacement

Question 2

A person walks 3 m east and then 4 m north. What is the magnitude of their total displacement?

Accepted Answer

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

Question 3

If you add two vectors graphically using the tip-to-tail method, what does the resultant vector represent?

Accepted Answer

The shortest path from the start of the first vector to the end of the last vector

Question 4

A box is pushed by two perpendicular forces: 12 N to the right and an unknown force upward. If the total force is 13 N, what is the magnitude of the upward force?

Accepted Answer

$$\sqrt{13^2$ - 12^2$} = 5 N$

Question 5

Which operation will change the magnitude but not the direction of a vector?

Accepted Answer

Multiplying the vector by a scalar

Question 6

Given $$\vec{A} = 8\hat{i} + 6\hat{j}$$, what is the magnitude of $$\vec{A}$$?

Accepted Answer

$$\sqrt{8^2$ + 6^2$} = 10$$

Question 7

A vector $$\vec{F} is 65 m $$long and directed 30.5° below the positive x-axis. What is the y-component $$F_y$$?

Accepted Answer

$$65 \sin(-30.5°)$$

Question 8

Which of the following statements about the dot product is correct?

Accepted Answer

The dot product of two perpendicular vectors is zero.

Question 9

Calculate the dot product $$\vec{A} \cdot \vec{B}$$ for $$\vec{A} = 2\hat{i} + 3\hat{j}$$ and $$\vec{B} = \hat{i} + 2\hat{j}.$$

Accepted Answer

$$2 	imes 1 + 3 	imes 2 = 8$$

Question 10

What is the result of the cross product $$\vec{A} 	imes \vec{B} if$$ $$\vec{A}$$ and $$\vec{B}$$ are parallel?

Accepted Answer

Zero

Question 11

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

Accepted Answer

$$|4 	imes 3 - 3 	imes (-2)| = 18$$

Question 12

A delivery truck travels 8 miles east, 5 miles north, and 4 miles east again. What is the magnitude of its final displacement from the origin?

Accepted Answer

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

Question 13

If $$\vec{A} = 10 m at$$ $$40°$$ above the +x-axis and $$\vec{B} = 3 m at$$ $$20°$$ above the x-axis, what is the x-component of $$\vec{A}$$?

Accepted Answer

$$10 \cos(40°)$$

Question 14

Which of the following best describes a unit vector?

Accepted Answer

A vector with magnitude 1 pointing in a specific direction

Question 15

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

Accepted Answer

$$2\hat{i} + 6\hat{j}$$

Question 16

A vector $$\vec{A}$$ has components $$A_x = -77 cm $$and $$A_y = 36 cm. $$What is the magnitude of $$\vec{A}$$?

Accepted Answer

$$\sqrt{(-77)^2$ + 36^2$} = 84.0 cm$$

Question 17

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 18

Two vectors $$\vec{A}$$ and $$\vec{B}$$ have a scalar product of $$-8$$ and a vector product of magnitude $$12. $$What is the angle between them?

Accepted Answer

$$\arccos\left(\frac{-8}{|A||B|}ight)$$ and $$\arcsin\left(\frac{12}{|A||B|}ight)$$, which are consistent for $$	heta = 123.7°$$

Vectors: Concepts, Operations, and Applications in Physics

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