Q: A vector $$\vec{A}$$ has components $$A_x = 3 m $$and $$A_y = 4 m. $$What is the direction angle $$\theta ($$relative to the x-axis)?

$$\theta = \tan$$^{-1}$$(4/3)$$

Q: A box slides down an incline. If you choose axes with +x downhill and +y perpendicular to the surface, what are the components of the gravitational acceleration $$\vec{a}_{\text{free fall}}$$?

$$a_x = g \sin \theta$$, $$a_y = g \cos \theta$$

Question 1

Which of the following is a vector quantity?

Accepted Answer

Velocity

Question 2

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

Accepted Answer

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

Question 3

Which of the following statements about vectors is correct?

Accepted Answer

Two vectors are equal if they have the same magnitude and direction.

Question 4

A bird flies 100 m east, then 50 m northwest (45° north of west). What is the x-component of the bird's net displacement?

Accepted Answer

$$100 - 50 \cos 45^\circ$$

Question 5

Which of the following best describes the result of multiplying a vector by a negative scalar?

Accepted Answer

The vector's magnitude is scaled and its direction reverses.

Question 6

Two vectors each have magnitude 4 units. What is the possible range of magnitudes for their sum?

Accepted Answer

0 to 8 units

Question 7

A hummingbird accelerates at $$6.0\ \mathrm{m/s^2$}$$, $$30^\circ$$ south of west. What are the x- and y-components of its acceleration if +x is east and +y is north?

Accepted Answer

$$a_x = -6.0 \cos 30^\circ$$, $$a_y = -6.0 \sin 30^\circ$$

Question 8

Which of the following is the correct unit vector form for a vector with components $$A_x$$ and $$A_y$$?

Accepted Answer

$$A_x \hat{\imath} + A_y \hat{\jmath}$$

Question 9

A rabbit runs at $$10.0\ \mathrm{m/s}$$, $$40.0^\circ$$ north of west. What is the x-component of its velocity if +x is east?

Accepted Answer

$$-10.0 \cos 40.0^\circ$$

Question 10

Which of the following expressions gives the magnitude of a vector $$\vec{C}$$ with components $$C_x$$ and $$C_y$$?

Accepted Answer

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

Question 11

If a vector has zero magnitude, what must be true about its components?

Accepted Answer

Every component must be zero.

Question 12

A vector $$\vec{A}$$ has components $$A_x = 3 m $$and $$A_y = 4 m. $$What is the direction angle $$	heta ($$relative to the x-axis)?

Accepted Answer

$$	heta = 	an$$^{-1}$$(4/3)$$

Question 13

Which of the following is true about the addition of vectors $$\vec{D}$$ and $$\vec{E}$$?

Accepted Answer

$$\vec{D} + \vec{E} = \vec{E} + \vec{D}$$

Question 14

A box slides down an incline. If you choose axes with +x downhill and +y perpendicular to the surface, what are the components of the gravitational acceleration $$\vec{a}_{	ext{free fall}}$$?

Accepted Answer

$$a_x = g \sin 	heta$$, $$a_y = g \cos 	heta$$

Question 15

Which of the following is NOT a property of unit vectors?

Accepted Answer

They have variable magnitude.

Vectors and Coordinate Systems: Foundations for Physics with Calculus

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