Q: A displacement vector $$\vec{r}$$ has components $$r_x = 0.997\ \mathrm{m}$$ and $$r_y = 0.465\ \mathrm{m}. $$What is the angle $$\theta it $$makes with the x-axis?

$$\theta = \tan$$^{-1}$$\left(\frac{0.465}{0.997}\right) = 25.0^\circ$$

Question 1

A vector $$\vec{A}$$ has a magnitude of $$5.0\ \mathrm{m}$$ and points at an angle of $$60.0^\circ$$ above the positive x-axis. What are the x- and y-components of $$\vec{A}$$?

Accepted Answer

$$A_x = 2.5\ \mathrm{m},\ A_y = 4.33\ \mathrm{m}$$

Question 2

Which of the following statements best distinguishes a vector from a scalar?

Accepted Answer

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

Question 3

A person walks 3 paces east, then 5 paces north. What is the magnitude of their net displacement?

Accepted Answer

$$\sqrt{3^2$ + 5^2$} = 5.83$$ paces

Question 4

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

Accepted Answer

$$\vec{C} = 2\hat{i} + 8\hat{j}$$

Question 5

A train moves east at $$15.0\ \mathrm{m/s}$$ and a passenger walks toward the front of the train at $$1.2\ \mathrm{m/s}$$ relative to the train. What is the passenger's velocity relative to the ground?

Accepted Answer

$$16.2\ \mathrm{m/s}$$ east

Question 6

Which equation correctly gives 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 7

A displacement vector $$\vec{r}$$ has components $$r_x = 0.997\ \mathrm{m}$$ and $$r_y = 0.465\ \mathrm{m}. $$What is the angle $$	heta it $$makes with the x-axis?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{0.465}{0.997}ight) = 25.0^\circ$$

Question 8

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

Accepted Answer

Unit vectors have magnitude 1 and no units.

Question 9

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

Accepted Answer

$$\sqrt{100^2$ + 100^2$} = 141\ \mathrm{m}$$

Question 10

If $$\vec{v}_{\mathrm{av}} = \frac{\Delta \vec{r}}{\Delta t}$$, which of the following best describes the direction of the average velocity vector?

Accepted Answer

It points in the same direction as the displacement vector.

Question 11

A boat heads due north across a river with a velocity of $$3.0\ \mathrm{m/s}$$ relative to the water. The river flows east at $$4.0\ \mathrm{m/s}. $$What is the magnitude of the boat's velocity relative to the ground?

Accepted Answer

$$5.0\ \mathrm{m/s}$$

Question 12

Which of the following is true about the acceleration vector?

Accepted Answer

It points in the direction of the change in velocity.

Question 13

A vector $$\vec{A} is $$multiplied by $$-3. $$What happens to its magnitude and direction?

Accepted Answer

The magnitude is tripled and the direction is reversed.

Question 14

A plane flies with an airspeed of $$200\ \mathrm{km/h}$$ due north. If a wind blows from the west at $$50\ \mathrm{km/h}$$, what is the plane's velocity relative to the ground?

Accepted Answer

$$\sqrt{200^2$ + 50^2$} = 206\ \mathrm{km/h}$$, 14^\circ$ east of north

Question 15

Which of the following is the correct formula for the y-component of a vector $$\vec{A} at an $$angle $$	heta$$ from the x-axis?

Accepted Answer

$$A_y = A \sin 	heta$$

Question 16

A displacement vector $$\vec{D}$$ points from the origin to the point $$(3, 4) in $$meters. What is the magnitude of $$\vec{D}$$?

Accepted Answer

$$5\ \mathrm{m}$$

Question 17

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

Accepted Answer

$$4\hat{i} - 1\hat{j}$$

Question 18

A runner moves with an average velocity $$\vec{v}_{\mathrm{av}}$$ for $$\Delta t = 10\ \mathrm{s}$$, covering a displacement $$\Delta \vec{r} = 50\ \mathrm{m}$$ east. What is the average velocity?

Accepted Answer

$$5.0\ \mathrm{m/s}$$ east

Vectors in Physics: Components, Operations, and Applications

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