Q: What is the angle of the hiker’s net displacement relative to east after walking 4 miles east and 3 miles north?

$$\tan$$^{-1}$$\left(\frac{3}{4}\right) = 37^\circ$$

Question 1

Which of the following best describes a scalar quantity?

Accepted Answer

A quantity that can be fully described by a number and units only.

Question 2

Sam and Bill both walk until they are 200 ft northeast of their respective front doors, but take different paths. What is true about their displacement vectors?

Accepted Answer

Their displacement vectors have the same magnitude and direction.

Question 3

A hiker walks 4 miles east and then 3 miles north. What is the magnitude of his net displacement?

Accepted Answer

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

Question 4

What is the angle of the hiker’s net displacement relative to east after walking 4 miles east and 3 miles north?

Accepted Answer

$$	an$$^{-1}$$\left(\frac{3}{4}ight) = 37^\circ$$

Question 5

Which method is best for obtaining precise answers when adding vectors?

Accepted Answer

Algebraic method

Question 6

If vector $$\vec{A}$$ has components $$A_x = 3 m $$and $$A_y = 2 m$$, what is the magnitude of $$\vec{A}$$?

Accepted Answer

$$\sqrt{3^2$ + 2^2$} = \sqrt{13} \approx 3.6 m$$

Question 7

Given a vector $$\vec{A}$$ with magnitude $$A$$ and angle $$	heta$$ from the x-axis, what is the x-component?

Accepted Answer

$$A_x = A \cos 	heta$$

Question 8

A vector $$\vec{B}$$ points down and to the right, and the angle $$\phi is $$adjacent to $$B_y. $$How do you find $$B_y$$?

Accepted Answer

$$B_y = -B \cos \phi$$

Question 9

Which of the following is a unit vector in the positive y-direction?

Accepted Answer

$$\hat{j}$$

Question 10

What is the vector sum $$\vec{A} + \vec{B} if$$ $$\vec{A} = 3\hat{i} + 2\hat{j}$$ and $$\vec{B} = -2\hat{i} + 4\hat{j}$$?

Accepted Answer

$$1\hat{i} + 6\hat{j}$$

Question 11

If a vector $$\vec{C} is $$decomposed into $$C_x$$ and $$C_y$$ components, which equation gives its magnitude?

Accepted Answer

$$C = \sqrt{C_x^2$ + C_y^2$}$$

Question 12

A vector $$\vec{A} is $$written as $$\vec{A} = A_x \hat{i} + A_y \hat{j}. $$What do $$\hat{i}$$ and $$\hat{j}$$ represent?

Accepted Answer

The directions of the x- and y-axes.

Question 13

Which of the following statements about the zero vector is correct?

Accepted Answer

It has zero length and no direction.

Question 14

If you multiply a vector $$\vec{A} by a $$negative scalar $$-c$$, what happens to its direction?

Accepted Answer

The direction reverses.

Question 15

A bird flies 100 m east and then 50 m northwest. Which method would be most precise for finding its net displacement?

Accepted Answer

Algebraic addition

Question 16

In a tilted coordinate system, what must be true for vector math to work?

Accepted Answer

The axes must be perpendicular to each other.

Question 17

A vector $$\vec{A}$$ has magnitude $$A$$ and points at angle $$	heta$$ above the x-axis. What is its y-component?

Accepted Answer

$$A_y = A \sin 	heta$$

Question 18

If a vector points left and down, what must you do when calculating its components?

Accepted Answer

Insert negative signs for both x and y components.

Question 19

Which of the following is NOT a reserved unit vector direction?

Accepted Answer

$$\hat{m}$$ for +m direction

Vectors and Coordinate Systems: Foundations and Applications

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