Q: Which of the following is the correct formula for the direction angle $$\theta of a $$vector $$\vec{A}$$ given its components $$A_x$$ and $$A_y$$?

$$\theta = \tan$$^{-1}$$ \left( \frac{A_y}{A_x} \right)$$

Question 1

Which of the following statements best describes the difference between a scalar and a vector quantity?

Accepted Answer

A vector has both magnitude and direction, while a scalar has only magnitude.

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

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

Question 3

A displacement vector $$\vec{r}$$ has a magnitude of $$1.10 m $$and makes an angle of $$25.0^\circ$$ with the positive $$x-$$axis. What are its $$x$$ and $$y$$ components?

Accepted Answer

$$r_x = 1.10 \cos 25.0^\circ$$, $$r_y = 1.10 \sin 25.0^\circ$$

Question 4

If $$\vec{A}$$ has components $$A_x = -5 m $$and $$A_y = 12 m$$, in which quadrant does $$\vec{A}$$ point?

Accepted Answer

Second quadrant

Question 5

Which of the following is the correct formula for the direction angle $$	heta of a $$vector $$\vec{A}$$ given its components $$A_x$$ and $$A_y$$?

Accepted Answer

$$	heta = 	an$$^{-1}$$ \left( \frac{A_y}{A_x} ight)$$

Question 6

A vector $$\vec{B}$$ has a magnitude of $$10 m $$and points $$60^\circ$$ above the positive $$x-$$axis. What is $$B_y$$?

Accepted Answer

$$B_y = 10 \sin 60^\circ = 8.66 m$$

Question 7

Which of the following statements about vector addition is correct?

Accepted Answer

Vector addition is commutative and associative.

Question 8

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

Accepted Answer

$$\vec{C} = 1\hat{x} + 9\hat{y}$$

Question 9

If $$\vec{D} = \vec{A} - \vec{B}$$, and $$\vec{A}$$ and $$\vec{B}$$ are given by $$\vec{A} = 4\hat{x} + 3\hat{y}$$ and $$\vec{B} = 1\hat{x} + 5\hat{y}$$, what is $$\vec{D}$$?

Accepted Answer

$$\vec{D} = 3\hat{x} - 2\hat{y}$$

Question 10

A vector $$\vec{A} is $$multiplied by a scalar $$-3. $$Which statement is true about $$-3\vec{A}$$?

Accepted Answer

The magnitude is three times larger and the direction is reversed.

Question 11

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 12

Which of the following is the correct unit vector notation for a vector with $$x-$$component $$6 m $$and $$y-$$component $$8 m$$?

Accepted Answer

$$6\hat{x} + 8\hat{y}$$

Question 13

A vector $$\vec{A}$$ has components $$A_x = 0$$ and $$A_y = -10 m. $$Which direction does $$\vec{A}$$ point?

Accepted Answer

Directly downward along the negative $$y-$$axis

Question 14

If $$\vec{A} = 5\hat{x} - 12\hat{y}$$, what is the magnitude of $$\vec{A}$$?

Accepted Answer

$$\sqrt{5^2$ + (-12)^2$} = 13$$

Question 15

A vector $$\vec{A} is $$described as $$\vec{A} = 7\hat{x} + 24\hat{y}. $$What is the angle $$	heta it $$makes with the positive $$x-$$axis?

Accepted Answer

$$	heta = 	an$$^{-1}$$ \left( \frac{24}{7} ight ) = 73.7^\circ$$

Question 16

Which of the following is a correct statement about unit vectors?

Accepted Answer

Unit vectors have a magnitude of 1 and indicate direction.

Question 17

A boat travels $$30 m $$east, then $$40 m $$north. What is the displacement vector in unit vector notation?

Accepted Answer

$$30\hat{x} + 40\hat{y}$$

Question 18

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

Accepted Answer

$$0\hat{x} + 0\hat{y}$$

Vectors in Physics: Definitions, Representation, and Operations

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