10 chioce questions of dot product of vectors

Here are 10 multiple-choice questions about the dot product of vectors:

Question 1:
What is the dot product of two vectors $\mathbf{a} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}$?
A) $\begin{pmatrix} 3 \\ -8 \end{pmatrix}$
B) $-5$
C) 11
D) $\begin{pmatrix} 4 \\ 2 \end{pmatrix}$
Correct Answer: B) $-5$

Question 2:
Given two vectors $\mathbf{u} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k}$ and $\mathbf{v} = \mathbf{i} + 2\mathbf{j} - 4\mathbf{k}$, what is $\mathbf{u} \cdot \mathbf{v}$?
A) $2\mathbf{i} - 6\mathbf{j} - 4\mathbf{k}$
B) $-8$
C) 12
D) $-10$
Correct Answer: B) $-8$

Question 3:
If the dot product of two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ is zero ($\mathbf{a} \cdot \mathbf{b} = 0$), what can be concluded about the relationship between $\mathbf{a}$ and $\mathbf{b}$?
A) They are parallel.
B) They are perpendicular (orthogonal).
C) They are equal in magnitude.
D) They point in the same direction.
Correct Answer: B) They are perpendicular (orthogonal).

Question 4:
Which of the following expressions represents the angle $\theta$ between two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$?
A) $\cos \theta = \frac{|\mathbf{a}| |\mathbf{b}|}{\mathbf{a} \cdot \mathbf{b}}$
B) $\sin \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}$
C) $\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}$
D) $\tan \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}$
Correct Answer: C) $\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}$

Question 5:
What is the dot product of a vector with itself, $\mathbf{a} \cdot \mathbf{a}$?
A) 0
B) $|\mathbf{a}|$
C) $|\mathbf{a}|^2$
D) $2|\mathbf{a}|$
Correct Answer: C) $|\mathbf{a}|^2$

Question 6:
If $\mathbf{a} = \begin{pmatrix} 2 \\ k \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -3 \\ 6 \end{pmatrix}$ are orthogonal, what is the value of $k$?
A) 1
B) $-1$
C) 2
D) $-2$
Correct Answer: A) 1

Question 7:
The scalar projection of vector $\mathbf{a}$ onto vector $\mathbf{b}$ is given by:
A) $\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|}$
B) $\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|}$
C) $\frac{|\mathbf{a}|}{|\mathbf{b}|}$
D) $\frac{|\mathbf{b}|}{|\mathbf{a}|}$
Correct Answer: B) $\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|}$

Question 8:
Given $|\mathbf{a}| = 4$, $|\mathbf{b}| = 3$, and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $60^\circ$, what is $\mathbf{a} \cdot \mathbf{b}$?
A) 12
B) 6
C) $6\sqrt{3}$
D) 0
Correct Answer: B) 6

Question 9:
Which of the following properties is true for the dot product?
A) $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = (\mathbf{a} \cdot \mathbf{b}) + \mathbf{c}$
B) $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$
C) $\mathbf{a} \cdot (\mathbf{b} \cdot \mathbf{c}) = (\mathbf{a} \cdot \mathbf{b}) \cdot \mathbf{c}$
D) $\mathbf{a} \cdot \mathbf{b}$ results in a vector.
Correct Answer: B) $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$

Question 10:
If $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are standard orthonormal basis vectors, what is $\mathbf{i} \cdot \mathbf{j}$?
A) 1
B) $-1$
C) 0
D) $\mathbf{k}$
Correct Answer: C) 0