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

Question 1

Which of the following statements best describes a vector quantity?

Accepted Answer

A quantity that has both magnitude and direction.

Question 2

A cross-country skier travels 1.00 km north and then 2.00 km east. What is the magnitude of her resultant displacement?

Accepted Answer

$$\sqrt{1.00^2$ + 2.00^2$} = 2.24 km$$

Question 3

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

Accepted Answer

$$\vec{A} \cdot \vec{B} = AB \cos \phi$$

Question 4

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

Accepted Answer

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

Question 5

Which of the following best describes the result of multiplying a vector $$\vec{A} by a $$negative scalar $$-3$$?

Accepted Answer

The new vector has three times the magnitude of $$\vec{A}$$ and points in the opposite direction.

Question 6

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

Accepted Answer

$$\hat{i} + 7\hat{j}$$

Question 7

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

Accepted Answer

Its direction is determined by the right-hand rule.

Question 8

If two vectors are perpendicular, what is their scalar (dot) product?

Accepted Answer

$$0$$

Question 9

A vector $$\vec{A}$$ makes an angle $$	heta$$ with the positive x-axis. What are its x- and y-components?

Accepted Answer

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

Question 10

Which of the following is a property of unit vectors?

Accepted Answer

They have a magnitude of 1 and no units.

Question 11

If $$\vec{A}$$ and $$\vec{B}$$ are parallel and point in the same direction, what is $$|\vec{A} + \vec{B}|$$?

Accepted Answer

$$|A + B|$$

Question 12

Which of the following best describes the graphical method for adding vectors?

Accepted Answer

Place the vectors head-to-tail and draw the resultant from the tail of the first to the head of the last.

Question 13

Given $$\vec{A} = 4\hat{i} - 3\hat{j}$$ and $$\vec{B} = -2\hat{i} + 5\hat{j}$$, what is the scalar product $$\vec{A} \cdot \vec{B}$$?

Accepted Answer

$$(4)(-2) + (-3)(5) = -8 - 15 = -23$$

Question 14

A force of $$10\ \mathrm{N}$$ acts at an angle of $$60^\circ to $$the horizontal. What is the horizontal component of the force?

Accepted Answer

$$10 \cos 60^\circ = 5\ \mathrm{N}$$

Question 15

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

Accepted Answer

It is anticommutative: $$\vec{A} 	imes \vec{B} = -\vec{B} 	imes \vec{A}.$$

Question 16

A vector $$\vec{A}$$ has components $$A_x = -3$$, $$A_y = 4. $$What is the angle $$	heta it $$makes with the positive x-axis?

Accepted Answer

$$	heta = \arctan\left(\frac{4}{-3}ight) = 126.87^\circ$$

Question 17

Which of the following is the correct formula for the magnitude of the vector sum $$\vec{R} = \vec{A} + \vec{B} in $$terms of components?

Accepted Answer

$$R = \sqrt{(A_x + B_x)^2$ + (A_y + B_y)^2$}$$

Question 18

If $$\vec{A}$$ and $$\vec{B}$$ are vectors in the $$xy-$$plane, which operation gives a vector perpendicular to both $$\vec{A}$$ and $$\vec{B}$$?

Accepted Answer

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

Question 19

A pilot flies 100 km east, then 100 km north. What is the magnitude and direction of the resultant displacement from the starting point?

Accepted Answer

Magnitude: $$\sqrt{100^2$ + 100^2$} = 141.4 km$$, Direction: $$45^\circ$$ north of east

Chapter 1: Units, Physical Quantities, and Vectors – Study Notes

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