Q: Given a vector $$\vec{A}$$ with magnitude $$A$$ and direction $$\theta$$ measured from the $$+x$$ axis, what are its $$x$$ and $$y$$ components?

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

Q: A vector $$\vec{B}$$ has components $$B_x = -3$$ and $$B_y = 4. $$What is the direction angle $$\theta (in $$degrees, measured counterclockwise from the $$+x$$ axis)?

$$\theta = \arctan\left(\frac{4}{-3}\right) = 126.9^\circ$$

Q: A vector $$\vec{E}$$ has components $$E_x = 3$$ and $$E_y = 4. $$What is the angle $$\theta it $$makes with the $$+x$$ axis?

$$\theta = \arctan\left(\frac{4}{3}\right) = 53.1^\circ$$

Question 1

Which of the following is a vector quantity?

Accepted Answer

Displacement

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

A vector has both magnitude and direction; a scalar has only magnitude.

Question 4

Given a vector $$\vec{A}$$ with magnitude $$A$$ and direction $$	heta$$ measured from the $$+x$$ axis, what are its $$x$$ and $$y$$ components?

Accepted Answer

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

Question 5

A velocity vector is given as $$\vec{v}_0 = (26.8, 22.5)\ \mathrm{m/s}. $$What is the magnitude of $$\vec{v}_0$$?

Accepted Answer

$$\sqrt{26.8^2$ + 22.5^2$}\ \mathrm{m/s}$$

Question 6

Which of the following quantities is NOT a vector?

Accepted Answer

Power

Question 7

A vector $$\vec{B}$$ has components $$B_x = -3$$ and $$B_y = 4. $$What is the direction angle $$	heta (in $$degrees, measured counterclockwise from the $$+x$$ axis)?

Accepted Answer

$$	heta = \arctan\left(\frac{4}{-3}ight) = 126.9^\circ$$

Question 8

Which of the following is a correct statement about the magnitude of a vector?

Accepted Answer

It is always positive.

Question 9

A gecko moves $$2.3\ \mathrm{m} at$$ $$250^\circ ($$trig direction). What are the $$x$$ and $$y$$ components of its displacement?

Accepted Answer

$$s_x = 2.3 \cos 250^\circ,\ s_y = 2.3 \sin 250^\circ$$

Question 10

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

Accepted Answer

$$\vec{A} = A_x \hat{i} + A_y \hat{j}$$

Question 11

A vector has a trigonometric direction of $$345^\circ. $$Which of the following is an equivalent direction?

Accepted Answer

$$-15^\circ$$

Question 12

If $$+x is $$East and $$+y is $$North, at which direction do the trigonometric angle and compass bearing have the same value?

Accepted Answer

$$0^\circ$$ and $$180^\circ$$

Question 13

Which of the following is a scalar quantity?

Accepted Answer

Power

Question 14

A vector $$\vec{C}$$ has components $$C_x = 0$$ and $$C_y = -5. $$What is the direction of $$\vec{C}$$?

Accepted Answer

$$270^\circ$$

Question 15

Which of the following is true about the components of a vector?

Accepted Answer

They can be either positive or negative.

Question 16

A vector $$\vec{D}$$ has a magnitude of $$10$$ units and points at $$120^\circ$$ from the $$+x$$ axis. What is $$D_x$$?

Accepted Answer

$$D_x = 10 \cos 120^\circ = -5$$

Question 17

Which of the following is the correct way to convert vector components to magnitude?

Accepted Answer

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

Question 18

A vector $$\vec{E}$$ has components $$E_x = 3$$ and $$E_y = 4. $$What is the angle $$	heta it $$makes with the $$+x$$ axis?

Accepted Answer

$$	heta = \arctan\left(\frac{4}{3}ight) = 53.1^\circ$$

Question 19

Which of the following is a correct statement about vectors?

Accepted Answer

Vectors can be moved anywhere in space as long as their magnitude and direction are unchanged.

Vectors, Scalars, and Vector Components: Foundations for Physics with Calculus

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