10 choice items of the dot product vector

Question 1: 1. If $\mathbf{a} = \langle 2, 3 \rangle$ and $\mathbf{b} = \langle -1, 4 \rangle$, what is $\mathbf{a} \cdot \mathbf{b}$?
A) 10
B) 14
C) -10
D) -14

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Question 2: 2. If $\mathbf{u} \cdot \mathbf{v} = 0$ for non-zero vectors $\mathbf{u}$ and $\mathbf{v}$, what can be concluded about the angle $\theta$ between them?
A) $\theta = 0^\circ$
B) $\theta = 45^\circ$
C) $\theta = 90^\circ$
D) $\theta = 180^\circ$

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Question 3: 3. Which of the following statements about the dot product is true?
A) $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$
B) $\mathbf{a} \cdot (\mathbf{b} \cdot \mathbf{c}) = (\mathbf{a} \cdot \mathbf{b}) \cdot \mathbf{c}$
C) $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = (\mathbf{a} \cdot \mathbf{b}) + \mathbf{c}$
D) $\mathbf{a} \cdot \mathbf{a} = 0$

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Question 4: 4. For any vector $\mathbf{v}$, what is $\mathbf{v} \cdot \mathbf{v}$ equal to?
A) $|\mathbf{v}|$
B) $2|\mathbf{v}|$
C) $|\mathbf{v}|^2$
D) 0

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Question 5: 5. 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) $\frac{|\mathbf{b}|}{|\mathbf{a}|}$

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Question 6: 6. What is the dot product of the standard unit vectors $\mathbf{i} \cdot \mathbf{j}$?
A) 1
B) -1
C) 0
D) $\mathbf{k}$

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Question 7: 7. If $|\mathbf{a}| = 3$, $|\mathbf{b}| = 4$, and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $60^\circ$, what is $\mathbf{a} \cdot \mathbf{b}$?
A) 6
B) $6\sqrt{3}$
C) 12
D) $12\sqrt{3}$

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Question 8: 8. For what value of $k$ are the vectors $\mathbf{u} = \langle 3, k \rangle$ and $\mathbf{v} = \langle -2, 6 \rangle$ orthogonal?
A) 1
B) -1
C) 2
D) -2

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Question 9: 9. If $\mathbf{a}$ is any vector and $\mathbf{0}$ is the zero vector, what is $\mathbf{a} \cdot \mathbf{0}$?
A) $|\mathbf{a}|$
B) 1
C) $\mathbf{a}$
D) 0

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Question 10: 10. If $\mathbf{a} \cdot \mathbf{b} < 0$, what can be said about the angle $\theta$ between vectors $\mathbf{a}$ and $\mathbf{b}$?
A) $\theta$ is acute ($0^\circ < \theta < 90^\circ$)
B) $\theta$ is obtuse ($90^\circ < \theta < 180^\circ$)
C) $\theta = 90^\circ$
D) $\theta = 0^\circ$ or $\theta = 180^\circ$