Q: A vector $$\vec{C}$$ has components $$C_x = -6 m $$and $$C_y = 8 m. $$What is the direction of $$\vec{C}$$ relative to the positive $$x-$$axis?

$$\theta = \tan$$^{-1}$$\left(\frac{8}{-6}\right) = -53.1^\circ ($$but in the second quadrant, so $$126.9^\circ)$$

Question 1

Which of the following statements best describes a vector quantity?

Accepted Answer

A physical quantity that has both magnitude and direction.

Question 2

Given a vector $$\vec{A}$$ with components $$A_x = 3 m $$and $$A_y = 4 m$$, what is the magnitude of $$\vec{A}$$?

Accepted Answer

$$5 m$$

Question 3

If a vector $$\vec{B}$$ has a magnitude of $$10 m $$and points at an angle of $$60^\circ$$ above the positive $$x-$$axis, what are its $$x$$ and $$y$$ components?

Accepted Answer

$$B_x = 10 \cos 60^\circ = 5 m$$, $$B_y = 10 \sin 60^\circ = 8.66 m$$

Question 4

A vector $$\vec{C}$$ has components $$C_x = -6 m $$and $$C_y = 8 m. $$What is the direction of $$\vec{C}$$ relative to the positive $$x-$$axis?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{8}{-6}ight) = -53.1^\circ ($$but in the second quadrant, so $$126.9^\circ)$$

Question 5

Which of the following is true about the addition of two vectors $$\vec{A}$$ and $$\vec{B}$$?

Accepted Answer

The sum $$\vec{A} + \vec{B} is $$independent of the order of addition.

Question 6

Given $$\vec{A} = 4\hat{x} + 3\hat{y}$$ and $$\vec{B} = -2\hat{x} + 5\hat{y}$$, what is $$\vec{C} = \vec{A} + \vec{B} in $$unit vector notation?

Accepted Answer

$$2\hat{x} + 8\hat{y}$$

Question 7

A vector $$\vec{D} is $$defined as $$\vec{D} = \vec{A} - \vec{B}. If$$ $$\vec{A} = 5\hat{x} + 2\hat{y}$$ and $$\vec{B} = 3\hat{x} - 4\hat{y}$$, what is $$\vec{D}$$?

Accepted Answer

$$2\hat{x} + 6\hat{y}$$

Question 8

If a vector $$\vec{A}$$ has a magnitude of $$7$$ units and points $$40^\circ$$ above the positive $$x-$$axis, what is the $$y-$$component of $$\vec{A}$$?

Accepted Answer

$$A_y = 7 \sin 40^\circ = 4.50$$ units

Question 9

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

Accepted Answer

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

Question 10

A displacement vector $$\vec{r}$$ has components $$r_x = 0.997 m $$and $$r_y = 0.465 m. $$What is the magnitude of $$\vec{r}$$?

Accepted Answer

$$r = \sqrt{(0.997)^2$ + (0.465)^2$} = 1.10 m$$

Question 11

If $$\vec{A}$$ has components $$A_x = -4$$ and $$A_y = -3$$, in which quadrant does $$\vec{A}$$ point?

Accepted Answer

Third quadrant

Question 12

Given two vectors $$\vec{A}$$ and $$\vec{B}$$, which of the following is the correct formula for the $$x-$$component of their sum $$\vec{C} = \vec{A} + \vec{B}$$?

Accepted Answer

$$C_x = A_x + B_x$$

Question 13

A vector $$\vec{A} is $$written in unit vector notation as $$\vec{A} = 6\hat{x} - 8\hat{y}. $$What is the magnitude of $$\vec{A}$$?

Accepted Answer

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

Question 14

If $$\vec{A} = 3\hat{x} + 4\hat{y}$$ and $$\vec{B} = 5\hat{x} - 2\hat{y}$$, what is $$\vec{A} - \vec{B}$$?

Accepted Answer

$$-2\hat{x} + 6\hat{y}$$

Question 15

A vector $$\vec{A}$$ has a magnitude of $$12$$ units and points $$150^\circ$$ counterclockwise from the positive $$x-$$axis. What is the $$x-$$component of $$\vec{A}$$?

Accepted Answer

$$A_x = 12 \cos 150^\circ = -10.39$$ units

Question 16

Which of the following is the correct way to express a two-dimensional vector $$\vec{A} in $$terms of unit vectors?

Accepted Answer

$$\vec{A} = A_x \hat{x} + A_y \hat{y}$$

Question 17

A rescue helicopter flies $$100 m $$east, then $$75 m $$north. What is the magnitude of its displacement from the starting point?

Accepted Answer

$$\sqrt{100^2$ + 75^2$} = 125 m$$

Question 18

If $$\vec{A} = 5\hat{x} + 12\hat{y}$$, what is the angle $$	heta$$ that $$\vec{A}$$ makes with the positive $$x-$$axis?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{12}{5}ight) = 67.4^\circ$$

Vectors in Physics: Definitions, Representations, and Operations

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