Q: Which equation correctly represents the dot product of two vectors $$\vec{A}$$ and $$\vec{B}$$?

$$\vec{A} \cdot \vec{B} = |A||B|\cos\theta$$

Q: If $$\vec{A} = a_1$\hat{i} + a_2$\hat{j} + a_3$\hat{k}$$ and $$\vec{B} = b_1$\hat{i} + b_2$\hat{j} + b_3$\hat{k}$$, what is the formula for their cross product $$\vec{A} \times \vec{B}$$?

$$\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ a_1$ & a_2$ & a_3$ \\ b_1$ & b_2$ & b_3$ \end{array}\right|$$

Question 1

Which of the following is a vector quantity?

Accepted Answer

Displacement

Question 2

A car travels 20 m/s to the northeast. Which physical quantity does this describe?

Accepted Answer

Velocity (vector)

Question 3

Which equation correctly represents the dot product of two vectors $$\vec{A}$$ and $$\vec{B}$$?

Accepted Answer

$$\vec{A} \cdot \vec{B} = |A||B|\cos	heta$$

Question 4

If a vector $$\vec{R}$$ has a magnitude of 10 units and makes an angle of $$30^\circ$$ with the x-axis, what is its x-component?

Accepted Answer

$$R_x = 10 \cos 30^\circ$$

Question 5

Which of the following statements about scalars and vectors is correct?

Accepted Answer

Scalars have only magnitude, while vectors have both magnitude and direction.

Question 6

A plane flies 215 km at $$22^\circ$$ east of due north. What is the eastward (x-axis) component of its displacement?

Accepted Answer

$$215 \sin 22^\circ$$

Question 7

Which law is illustrated by the equation $$\vec{a} + \vec{b} = \vec{b} + \vec{a}$$?

Accepted Answer

Commutative law

Question 8

What is the result of multiplying a vector by a scalar?

Accepted Answer

The magnitude changes but the direction remains the same.

Question 9

Given $$\vec{a} = 2\hat{i} + 2\hat{j} - 5\hat{k}$$ and $$\vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}$$, what is the unit vector in the direction of $$\vec{a} + \vec{b}$$?

Accepted Answer

$$\frac{1}{\sqrt{29}}(4\hat{i} + 3\hat{j} - 2\hat{k})$$

Question 10

Which of the following is NOT a vector quantity?

Accepted Answer

Density

Question 11

If $$\vec{A} = 3\hat{i} + 4\hat{j}$$ and $$\vec{B} = 5\hat{i} - 2\hat{j}$$, what is $$\vec{A} + \vec{B}$$?

Accepted Answer

$$8\hat{i} + 2\hat{j}$$

Question 12

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

Accepted Answer

It results in a vector quantity.

Question 13

What is the dot product of two perpendicular vectors?

Accepted Answer

It is zero.

Question 14

Which of the following is the correct formula for the magnitude of a vector $$\vec{a} = a_x \hat{i} + a_y \hat{j}$$?

Accepted Answer

$$|\vec{a}| = \sqrt{a_x^2$ + a_y^2$}$$

Question 15

A vector $$\vec{A} is $$multiplied by another vector $$\vec{B}$$ using the cross product. Which of the following is true?

Accepted Answer

The result is a vector perpendicular to both $$\vec{A}$$ and $$\vec{B}.$$

Question 16

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

Accepted Answer

Unit vectors have magnitude equal to 1 and point in a specific direction.

Question 17

If $$\vec{A} = a_1$\hat{i} + a_2$\hat{j} + a_3$\hat{k}$$ and $$\vec{B} = b_1$\hat{i} + b_2$\hat{j} + b_3$\hat{k}$$, what is the formula for their cross product $$\vec{A} 	imes \vec{B}$$?

Accepted Answer

$$\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \ a_1$ & a_2$ & a_3$ \ b_1$ & b_2$ & b_3$ \end{array}ight|$$

Question 18

Which of the following is NOT a property of the dot product?

Accepted Answer

It results in a vector perpendicular to the original vectors.

Question 19

A hiker walks 10 km north, then 5 km west. What is the magnitude of the resultant displacement?

Accepted Answer

$$\sqrt{10^2$ + 5^2$} km$$

Vectors and Scalars: Foundations and Operations in Physics

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