JEE (Main) Mathematics - VECTOR ALGEBRA MCQs (61-75)
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Which of the following represents a vector?
- a) Speed
- b) Temperature
- c) Force
- d) Energy
Answer: c) Force
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If \( \mathbf{A} = 3\hat{i} - 4\hat{j} + 5\hat{k} \) and \( \mathbf{B} = \hat{i} + 2\hat{j} - 3\hat{k} \), the dot product \( \mathbf{A} \cdot \mathbf{B} \) is:
- a) 3
- b) 4
- c) 0
- d) -10
Answer: d) -10
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The magnitude of a vector \( \mathbf{A} = 4\hat{i} - 3\hat{j} \) is:
- a) 5
- b) 7
- c) 4
- d) 3
Answer: a) 5
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The direction cosines of a vector \( \mathbf{A} = 3\hat{i} + 4\hat{j} + 12\hat{k} \) are:
- a) \( \frac{1}{\sqrt{2}}, \frac{2}{\sqrt{2}}, \frac{3}{\sqrt{2}} \)
- b) \( \frac{1}{3}, \frac{4}{3}, \frac{12}{3} \)
- c) \( \frac{3}{\sqrt{169}}, \frac{4}{\sqrt{169}}, \frac{12}{\sqrt{169}} \)
- d) \( \frac{3}{\sqrt{9}}, \frac{4}{\sqrt{9}}, \frac{12}{\sqrt{9}} \)
Answer: c) \( \frac{3}{\sqrt{169}}, \frac{4}{\sqrt{169}}, \frac{12}{\sqrt{169}} \)
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If two vectors \( \mathbf{A} = 4\hat{i} + 3\hat{j} \) and \( \mathbf{B} = 2\hat{i} + 5\hat{j} \), the sum of the vectors \( \mathbf{A} + \mathbf{B} \) is:
- a) \( 6\hat{i} + 8\hat{j} \)
- b) \( 6\hat{i} + 3\hat{j} \)
- c) \( 2\hat{i} + 5\hat{j} \)
- d) \( 8\hat{i} + 6\hat{j} \)
Answer: a) \( 6\hat{i} + 8\hat{j} \)
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The vector equation of a line passing through the origin in the direction of vector \( \mathbf{A} = 3\hat{i} + 4\hat{j} \) is:
- a) \( \mathbf{r} = 3\hat{i} + 4\hat{j} \)
- b) \( \mathbf{r} = t(3\hat{i} + 4\hat{j}) \)
- c) \( \mathbf{r} = 3t\hat{i} + 4t\hat{j} \)
- d) \( \mathbf{r} = \hat{i} + \hat{j} \)
Answer: b) \( \mathbf{r} = t(3\hat{i} + 4\hat{j}) \)
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The component of vector \( \mathbf{A} = 4\hat{i} + 3\hat{j} + 2\hat{k} \) along the direction of \( \hat{i} \) is:
- a) 4
- b) 3
- c) 2
- d) 5
Answer: a) 4
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The cross product of two parallel vectors \( \mathbf{A} \) and \( \mathbf{B} \) is:
- a) Zero
- b) Non-zero vector
- c) Scalar
- d) Undefined
Answer: a) Zero
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If the dot product of two vectors \( \mathbf{A} \cdot \mathbf{B} = 0 \), then the vectors \( \mathbf{A} \) and \( \mathbf{B} \) are:
- a) Parallel
- b) Perpendicular
- c) Collinear
- d) None of the above
Answer: b) Perpendicular
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The magnitude of the vector \( \mathbf{A} = 2\hat{i} - 3\hat{j} + \hat{k} \) is:
- a) \( \sqrt{14} \)
- b) \( \sqrt{12} \)
- c) \( \sqrt{10} \)
- d) \( \sqrt{9} \)
Answer: a) \( \sqrt{14} \)
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If \( \mathbf{A} = 4\hat{i} + 2\hat{j} \) and \( \mathbf{B} = -2\hat{i} + 3\hat{j} \), the scalar product \( \mathbf{A} \cdot \mathbf{B} \) is:
- a) 0
- b) 2
- c) -2
- d) 10
Answer: b) 2
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The angle between the vectors \( \mathbf{A} = 3\hat{i} + \hat{j} \) and \( \mathbf{B} = 2\hat{i} + 4\hat{j} \) is:
- a) \( 0^\circ \)
- b) \( 45^\circ \)
- c) \( 60^\circ \)
- d) \( 90^\circ \)
Answer: c) \( 60^\circ \)
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The area of a parallelogram formed by two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by:
- a) \( |\mathbf{A} \times \mathbf{B}| \)
- b) \( |\mathbf{A} \cdot \mathbf{B}| \)
- c) \( |\mathbf{A} + \mathbf{B}| \)
- d) \( |\mathbf{A}| |\mathbf{B}| \)
Answer: a) \( |\mathbf{A} \times \mathbf{B}| \)
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The projection of a vector \( \mathbf{A} \) on vector \( \mathbf{B} \) is given by:
- a) \( \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|} \)
- b) \( \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}|} \)
- c) \( \mathbf{A} \times \mathbf{B} \)
- d) None of the above
Answer: a) \( \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|} \)
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