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

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

Q: If $$\vec{A} = 3\hat{i} + 4\hat{j}$$, what is the direction angle $$\theta ($$measured from the positive x-axis)?

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

Q: A vector in the xy-plane has components $$A_x = -4$$ and $$A_y = 3. $$What is the angle it makes with the positive x-axis?

$$180^\circ + \tan$$^{-1}$$(3/4)$$

Question 1

Which of the following best describes a vector quantity?

Accepted Answer

A quantity that has both magnitude and direction.

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

$$5$$ miles

Question 4

Which method is NOT a valid graphical technique for vector addition?

Accepted Answer

Scalar multiplication method

Question 5

If a vector $$\vec{A} is $$multiplied by a negative scalar $$-2$$, what happens to the vector?

Accepted Answer

Its magnitude is doubled and direction is reversed.

Question 6

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

Accepted Answer

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

Question 7

Which of the following statements about vector addition is correct?

Accepted Answer

Vector addition is commutative and associative.

Question 8

A ball is tossed straight up in the air. What is the direction of the acceleration vector during its flight (ignoring air resistance)?

Accepted Answer

Points down when moving up and down.

Question 9

Given vector components $$A_x = 3$$ and $$A_y = 4$$, what is the magnitude of vector $$\vec{A}$$?

Accepted Answer

$$5$$

Question 10

A force of $$10\ \mathrm{N} is $$applied horizontally to a surface tilted at $$20^\circ. $$What is the component of the force perpendicular to the surface?

Accepted Answer

$$10 \sin 20^\circ$$

Question 11

Which of the following is true about unit vectors?

Accepted Answer

They have magnitude 1 and no units.

Question 12

If $$\vec{A} = 3\hat{i} + 4\hat{j}$$, what is the direction angle $$	heta ($$measured from the positive x-axis)?

Accepted Answer

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

Question 13

A bird flies $$100\ \mathrm{m}$$ due east, then $$200\ \mathrm{m}$$ northwest ($$45^\circ$$ west of north). What is the x-component of the bird's net displacement?

Accepted Answer

$$100 - 200 \cos 45^\circ$$

Question 14

Which of the following is the correct way to subtract vector $$\vec{B}$$ from vector $$\vec{A}$$ graphically?

Accepted Answer

Add $$-\vec{B} to$$ $$\vec{A}$$ using the head-to-tail method.

Question 15

Given a vector $$\vec{C}$$ with $$C_x = -3$$ and $$C_y = 4$$, what is the magnitude of $$\vec{C}$$?

Accepted Answer

$$5$$

Question 16

Which of the following is true about the displacement vector?

Accepted Answer

It depends only on the initial and final positions.

Question 17

A vector in the xy-plane has components $$A_x = -4$$ and $$A_y = 3. $$What is the angle it makes with the positive x-axis?

Accepted Answer

$$180^\circ + 	an$$^{-1}$$(3/4)$$

Question 18

Which of the following is a property of vector addition?

Accepted Answer

$$\vec{A} + \vec{B} = \vec{B} + \vec{A}$$

Question 19

If a vector $$\vec{A} is $$decomposed into components parallel to the axes, what does this mean?

Accepted Answer

It is broken into pieces parallel to the axes.

Vectors and Coordinate Systems: Foundations and Applications

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