Q: Given $$ \vec{A} = A_x \hat{i} + A_y \hat{j} $$, what is the direction angle $$ \theta $$ relative to the x-axis?

$$\theta = \tan$$^{-1}$$ \left( \frac{A_y}{A_x} \right)$$

Question 1

A vector $$ \vec{A} $$ has a magnitude of 5 units and points at an angle of 30° above the positive x-axis. What are the x and y components of $$ \vec{A} $$?

Accepted Answer

$$A_x = 5 \cos 30^\circ = 4.33$$, $$A_y = 5 \sin 30^\circ = 2.5$$

Question 2

Which statement best describes a unit vector?

Accepted Answer

A unit vector has a magnitude of 1 and points in a specific direction.

Question 3

Carolyn drives her car north at 30 km/hr for 1 hour, east at 60 km/hr for 2 hours, then north at 50 km/hr for 1 hour. What is her net displacement?

Accepted Answer

$$\vec{D} = 120 \hat{i} + 80 \hat{j} km$$

Question 4

Given $$ \vec{B} $$ has components $$B_x = -2 m $$and $$B_y = 2 m$$, what is the magnitude of $$ \vec{B} $$?

Accepted Answer

$$|\vec{B}| = \sqrt{(-2)^2$ + (2)^2$} = 2.83 m$$

Question 5

A rabbit runs 40° north of west at 10 m/s. What are the velocity components in the x (east) and y (north) directions?

Accepted Answer

$$v_x = -10 \cos 40^\circ$$, $$v_y = 10 \sin 40^\circ$$

Question 6

If $$ \vec{A} = 3 \hat{i} + 4 \hat{j} $$, what is the magnitude of $$ \vec{A} $$?

Accepted Answer

$$|\vec{A}| = \sqrt{3^2$ + 4^2$} = 5$$

Question 7

Which rule is used to add two vectors graphically by placing the tail of one at the tip of the other?

Accepted Answer

Tip-to-tail rule

Question 8

A vector $$ \vec{A}  is $$multiplied by a scalar $$c = 3. $$What happens to $$ \vec{A} $$?

Accepted Answer

The magnitude triples, but the direction remains the same.

Question 9

If $$ \vec{A} $$ points down and left, which signs should its x and y components have?

Accepted Answer

$$A_x \u003c 0$$, $$A_y \u003c 0$$

Question 10

Given $$ \vec{A} = A_x \hat{i} + A_y \hat{j} $$, what is the direction angle $$ 	heta $$ relative to the x-axis?

Accepted Answer

$$	heta = 	an$$^{-1}$$ \left( \frac{A_y}{A_x} ight)$$

Question 11

Which of the following is NOT a property of vectors?

Accepted Answer

Vectors can be multiplied to yield a scalar with direction.

Question 12

A vector $$ \vec{C}  is $$given by $$ \vec{C} = 2\vec{A} + \vec{B} . If$$ $$ \vec{A} = 1 \hat{i} + 2 \hat{j} $$ and $$ \vec{B} = 3 \hat{i} + 1 \hat{j} $$, what is $$ \vec{C} $$?

Accepted Answer

$$\vec{C} = 5 \hat{i} + 5 \hat{j}$$

Question 13

If a vector $$ \vec{A} $$ has components $$A_x = 0$$ and $$A_y = -5$$, which direction does it point?

Accepted Answer

Negative y-direction

Question 14

Which formula gives the magnitude of a vector $$ \vec{A} $$ with components $$A_x$$ and $$A_y$$?

Accepted Answer

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

Question 15

A vector $$ \vec{A}  is $$decomposed as $$\vec{A} = A_x \hat{i} + A_y \hat{j}. $$What is the physical meaning of $$\hat{i}$$ and $$\hat{j}$$?

Accepted Answer

$$\hat{i}$$ and $$\hat{j}$$ are unit vectors in the x and y directions, respectively.

Question 16

If $$ \vec{A} $$ points in the negative x-direction, what is the sign of $$A_x$$?

Accepted Answer

Negative

Question 17

Which method is used to add two vectors by constructing a parallelogram?

Accepted Answer

Parallelogram rule

Question 18

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

Accepted Answer

$$	heta = 	an$$^{-1}$$ \left( \frac{4}{3} ight) \approx 53.1^\circ$$

Question 19

If $$ \vec{A} $$ and $$ \vec{B} $$ are vectors, which equation represents their sum in component form?

Accepted Answer

$$\vec{A} + \vec{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j}$$

Chapter 3: Vectors and Coordinate Systems – Structured Study Notes

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