Q: Which of the following best describes the dot product of two vectors $$\vec{a}$$ and $$\vec{b}$$?

It is a scalar given by $$\vec{a} \cdot \vec{b} = ab \cos \theta$$

Question 1

Which of the following statements best distinguishes a vector from a scalar quantity?

Accepted Answer

A vector has both magnitude and direction, while a scalar has only magnitude.

Question 2

Given a vector $$\vec{b}$$ with components $$b_x = 7\ \mathrm{m}$$ and $$b_y = -5\ \mathrm{m}$$, what is the magnitude of $$\vec{b}$$?

Accepted Answer

$$\sqrt{7^2$ + (-5)^2$} = \sqrt{74} \approx 8.6\ \mathrm{m}$$

Question 3

If a vector $$\vec{a}$$ makes an angle $$	heta$$ with the positive x-axis, which formula gives the x-component of $$\vec{a}$$?

Accepted Answer

$$a_x = a \cos 	heta$$

Question 4

Which of the following is a correct statement about the addition of vectors?

Accepted Answer

Vector addition is commutative: $$\vec{a} + \vec{b} = \vec{b} + \vec{a}$$

Question 5

A car moves from position $$x_1 = -3$$\ \mathrm{m}$$ to x_2$ = 7\ \mathrm{m}. $$What is its displacement?

Accepted Answer

$$7\ \mathrm{m} - (-3\ \mathrm{m}) = 10\ \mathrm{m}$$

Question 6

Which of the following best describes the dot product of two vectors $$\vec{a}$$ and $$\vec{b}$$?

Accepted Answer

It is a scalar given by $$\vec{a} \cdot \vec{b} = ab \cos 	heta$$

Question 7

If the dot product $$\vec{C} \cdot \vec{D} = 0$$ and $$|\vec{C}| = 3$$, $$|\vec{D}| = 4$$, what is the angle between $$\vec{C}$$ and $$\vec{D}$$?

Accepted Answer

$$90^\circ$$

Question 8

A ball is thrown straight up. At the highest point of its trajectory, what is the acceleration of the ball (neglecting air resistance)?

Accepted Answer

$$-9.8\ \mathrm{m/s^2$}$$ downward

Question 9

A car with an initial velocity of $$v_0 = -25$$\ \mathrm{m/s}$$ is brought to rest in 5.0$$\ \mathrm{s}$$. What is its acceleration?

Accepted Answer

$$a = \frac{0 - (-25)}{5.0} = 5.0\ \mathrm{m/s^2$}$$

Question 10

Which of the following is true about the cross product $$\vec{a} 	imes \vec{b}$$?

Accepted Answer

It results in a vector perpendicular to both $$\vec{a}$$ and $$\vec{b}.$$

Question 11

A cyclist travels 400 m east from a starting point, then 400 m west. What is the net displacement?

Accepted Answer

$$0\ \mathrm{m}$$

Question 12

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

Accepted Answer

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

Question 13

Which of the following is a correct statement about unit vectors?

Accepted Answer

Unit vectors have a magnitude of 1 and point along coordinate axes.

Question 14

A vector $$\vec{v} is $$multiplied by a scalar $$c = -2. $$What happens to the direction and magnitude of the resulting vector?

Accepted Answer

The magnitude is doubled and the direction is reversed.

Question 15

If a car starts from rest and accelerates at $$2\ \mathrm{m/s^2$}$$ for $$5\ \mathrm{s}$$, what is its final velocity?

Accepted Answer

$$v = 0 + (2)(5) = 10\ \mathrm{m/s}$$

Question 16

Which of the following is the correct formula for the position $$x of an $$object under constant acceleration $$a$$ after time $$t$$, given initial position $$x_0$$ and initial velocity $$v_0$$?

Accepted Answer

$$x = x_0$ + v_0$ t + \frac{1}{2} a t^2$

Question 17

If two vectors $$\vec{a}$$ and $$\vec{b}$$ are perpendicular, what is the value of their dot product?

Accepted Answer

$$0$$

Question 18

A runner moves with a constant velocity of $$5\ \mathrm{m/s}$$ for $$10\ \mathrm{s}. $$What is the total displacement?

Accepted Answer

$$5 	imes 10 = 50\ \mathrm{m}$$

Vectors and Kinematics: Foundations for Physics with Calculus

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