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

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

Q: A car travels 100 m east, then 100 m north. What is the magnitude of its displacement?

$$\sqrt{100^2$ + 100^2$} = 141.4\ \text{m}$$

Q: Convert $$20\ \text{mm}^3 to$$ $$\text{m}^3.$$

$$20 \times 10$$^{-9}$$\ \text{m}^3$$

Q: If $$\vec{A} = 6\hat{i}$$ and $$\vec{B}$$ has a magnitude of 4 units and lies in the y-z plane, making an angle of $$30^\circ$$ with the x-axis, what is the magnitude of $$\vec{A} \times \vec{B}$$?

$$6 \times 4 \times \sin(30^\circ) = 12$$

Question 1

Which of the following statements about significant figures is correct according to the rules presented?

Accepted Answer

Trailing zeros in a decimal number are significant.

Question 2

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

Accepted Answer

$$A_x = 10 \cos(30^\circ),\ A_y = 10 \sin(30^\circ)$$

Question 3

Given vectors $$\vec{A} = (3, 5)$$ and $$\vec{B} = (4, 2)$$, what is the magnitude of $$\vec{C} = \vec{A} + \vec{B}$$?

Accepted Answer

$$\sqrt{(3+4)^2$ + (5+2)^2$} = \sqrt{49 + 49} = 9.90$$

Question 4

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

Accepted Answer

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

Question 5

If $$\vec{A} = 2\hat{i} + 3\hat{j} + 1\hat{k}$$ and $$\vec{B} = -4\hat{i} + 2\hat{j} - 1\hat{k}$$, what is $$\vec{A} \cdot \vec{B}$$?

Accepted Answer

$$-8 + 6 - 1 = -3$$

Question 6

Which of the following best describes the right-hand rule for the cross product $$\vec{A} 	imes \vec{B}$$?

Accepted Answer

Curl the fingers of your right hand from $$\vec{A} to$$ $$\vec{B}$$; your thumb points in the direction of $$\vec{A} 	imes \vec{B}.$$

Question 7

A car travels 100 m east, then 100 m north. What is the magnitude of its displacement?

Accepted Answer

$$\sqrt{100^2$ + 100^2$} = 141.4\ 	ext{m}$$

Question 8

Which of the following is NOT a correct property of unit vectors in Cartesian coordinates?

Accepted Answer

The cross product of any two unit vectors is zero.

Question 9

Convert $$20\ 	ext{mm}^3 to$$ $$	ext{m}^3.$$

Accepted Answer

$$20 	imes 10$$^{-9}$$\ 	ext{m}^3$$

Question 10

If $$\vec{A} = 6\hat{i}$$ and $$\vec{B}$$ has a magnitude of 4 units and lies in the y-z plane, making an angle of $$30^\circ$$ with the x-axis, what is the magnitude of $$\vec{A} 	imes \vec{B}$$?

Accepted Answer

$$6 	imes 4 	imes \sin(30^\circ) = 12$$

Question 11

Which of the following best describes the result of the scalar product $$\vec{A} \cdot \vec{B}$$ when the angle between $$\vec{A}$$ and $$\vec{B} is$$ $$90^\circ$$?

Accepted Answer

It is zero.

Question 12

A vector $$\vec{R}$$ has components $$R_x = 4$$ and $$R_y = 3. $$What is the direction of $$\vec{R}$$ with respect to the x-axis?

Accepted Answer

$$	an$$^{-1}$$(3/4)$$

Question 13

Which of the following is the correct expression for the magnitude of a vector $$\vec{A}$$ with components $$A_x$$ and $$A_y$$?

Accepted Answer

$$A = \sqrt{A_x^2$ + A_y^2$}$$

Question 14

Which of the following is a correct property of the cross product of two vectors?

Accepted Answer

The cross product is anticommutative.

Question 15

A ball is thrown straight up with an initial velocity of $$20\ 	ext{m/s}. $$Neglecting air resistance, what is its velocity after $$2\ 	ext{s}$$? (Take $$g = 9.8\ 	ext{m/s}^2$$ downward.)

Accepted Answer

$$0.4\ 	ext{m/s}$$

Question 16

Which of the following is the correct formula for average velocity $$\bar{v}$$ over a time interval $$\Delta t$$?

Accepted Answer

$$\bar{v} = \frac{\Delta x}{\Delta t}$$

Question 17

If a vector $$\vec{A} is $$described as $$\vec{A} = 4\hat{i} - 3\hat{j}$$, what is its magnitude?

Accepted Answer

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

Question 18

Which of the following is a correct statement about the sum of two vectors $$\vec{A}$$ and $$\vec{B}$$?

Accepted Answer

The direction of $$\vec{A} + \vec{B}$$ depends on both the magnitudes and directions of $$\vec{A}$$ and $$\vec{B}.$$

Units, Physical Quantities, and Vectors: Structured Study Notes

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