JEE (Main) Mathematics - VECTOR ALGEBRA Multiple Choice Questions (MCQs)

JEE (Main) Mathematics - Vectors and Scalars MCQs

1. Q1. Which of the following is a scalar quantity?
   a) Displacement
   b) Velocity
   c) Force
   d) Temperature
   Answer: d) Temperature
2. Q2. The vector \( \mathbf{A} \) has a magnitude of 5 units and makes an angle of 30° with the positive x-axis. What is the x-component of \( \mathbf{A} \)?
   a) \( 5 \)
   b) \( 5\sqrt{3} \)
   c) \( 5 \cdot \cos(30^\circ) \)
   d) \( 5 \cdot \sin(30^\circ) \)
   Answer: c) \( 5 \cdot \cos(30^\circ) \)
3. Q3. Which of the following operations is NOT valid for vectors?
   a) Addition
   b) Subtraction
   c) Scalar multiplication
   d) Scalar division
   Answer: d) Scalar division
4. Q4. If \( \mathbf{A} = 3\hat{i} + 2\hat{j} \) and \( \mathbf{B} = 4\hat{i} - \hat{j} \), then the sum of \( \mathbf{A} \) and \( \mathbf{B} \) is:
   a) \( \mathbf{A} + \mathbf{B} = 7\hat{i} + \hat{j} \)
   b) \( \mathbf{A} + \mathbf{B} = 7\hat{i} - \hat{j} \)
   c) \( \mathbf{A} + \mathbf{B} = 7\hat{i} + 3\hat{j} \)
   d) \( \mathbf{A} + \mathbf{B} = 3\hat{i} + 3\hat{j} \)
   Answer: c) \( \mathbf{A} + \mathbf{B} = 7\hat{i} + 3\hat{j} \)
5. Q5. The magnitude of a vector \( \mathbf{A} = 4\hat{i} + 3\hat{j} + 12\hat{k} \) is:
   a) \( \sqrt{9 + 16 + 144} \)
   b) \( \sqrt{16 + 9 + 144} \)
   c) \( \sqrt{16 + 9 + 12} \)
   d) \( \sqrt{16 + 3 + 12} \)
   Answer: b) \( \sqrt{16 + 9 + 144} \)
6. Q6. The dot product of two vectors \( \mathbf{A} = 3\hat{i} + 4\hat{j} \) and \( \mathbf{B} = 2\hat{i} - 3\hat{j} \) is:
   a) \( 6 \)
   b) \( -6 \)
   c) \( 12 \)
   d) \( -12 \)
   Answer: b) \( -6 \)
7. Q7. If the vectors \( \mathbf{A} = 2\hat{i} + 3\hat{j} + \hat{k} \) and \( \mathbf{B} = \hat{i} - 2\hat{j} + 4\hat{k} \), then their cross product is:
   a) \( 5\hat{i} - 7\hat{j} - 7\hat{k} \)
   b) \( 5\hat{i} + 7\hat{j} - 7\hat{k} \)
   c) \( 5\hat{i} + 7\hat{j} + 7\hat{k} \)
   d) \( 5\hat{i} - 7\hat{j} + 7\hat{k} \)
   Answer: b) \( 5\hat{i} + 7\hat{j} - 7\hat{k} \)
8. Q8. The vector \( \mathbf{A} \) is along the x-axis and the vector \( \mathbf{B} \) is along the y-axis. What is the value of \( \mathbf{A} \cdot \mathbf{B} \)?
   a) 1
   b) 0
   c) -1
   d) Undefined
   Answer: b) 0
9. Q9. The unit vector in the direction of \( \mathbf{A} = 3\hat{i} + 4\hat{j} \) is:
   a) \( \frac{3\hat{i} + 4\hat{j}}{5} \)
   b) \( \frac{3\hat{i} + 4\hat{j}}{7} \)
   c) \( \frac{4\hat{i} + 3\hat{j}}{7} \)
   d) \( \frac{5\hat{i} + 4\hat{j}}{3} \)
   Answer: a) \( \frac{3\hat{i} + 4\hat{j}}{5} \)
10. Q10. If the vectors \( \mathbf{A} = 2\hat{i} + 3\hat{j} \) and \( \mathbf{B} = \hat{i} + 4\hat{j} \), the angle between \( \mathbf{A} \) and \( \mathbf{B} \) is:
    a) \( 60^\circ \)
    b) \( 45^\circ \)
    c) \( 90^\circ \)
    d) \( 30^\circ \)
    Answer: b) \( 45^\circ \)
11. Q11. Which of the following represents the vector projection of \( \mathbf{A} \) onto \( \mathbf{B} \)?
    a) \( \frac{\mathbf{A} \cdot \mathbf{B}}{\mathbf{B} \cdot \mathbf{B}} \mathbf{B} \)
    b) \( \mathbf{A} \times \mathbf{B} \)
    c) \( \mathbf{A} + \mathbf{B} \)
    d) \( \frac{\mathbf{A} \times \mathbf{B}}{\mathbf{A} \cdot \mathbf{B}} \)
    Answer: a) \( \frac{\mathbf{A} \cdot \mathbf{B}}{\mathbf{B} \cdot \mathbf{B}} \mathbf{B} \)
12. Q12. The magnitude of the cross product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is equal to:
    a) \( |\mathbf{A}| |\mathbf{B}| \)
    b) \( |\mathbf{A}| |\mathbf{B}| \cos(\theta) \)
    c) \( |\mathbf{A}| |\mathbf{B}| \sin(\theta) \)
    d) \( |\mathbf{A}| + |\mathbf{B}| \)
    Answer: c) \( |\mathbf{A}| |\mathbf{B}| \sin(\theta) \)
13. Q13. If \( \mathbf{A} = \hat{i} + 2\hat{j} \) and \( \mathbf{B} = 3\hat{i} + \hat{j} \), then \( \mathbf{A} \times \mathbf{B} \) is:
    a) \( \hat{k} \)
    b) \( -\hat{k} \)
    c) \( 3\hat{k} \)
    d) \( 3\hat{i} \)
    Answer: a) \( \hat{k} \)
14. Q14. If \( \mathbf{A} = \hat{i} + 2\hat{j} \) and \( \mathbf{B} = 3\hat{i} + \hat{j} \), the dot product of \( \mathbf{A} \) and \( \mathbf{B} \) is:
    a) 5
    b) 7
    c) 9
    d) 6
    Answer: b) 7
15. Q15. Which of the following vectors is perpendicular to both \( \mathbf{A} = \hat{i} + 2\hat{j} \) and \( \mathbf{B} = 3\hat{i} + \hat{j} \)?
    a) \( \hat{i} \)
    b) \( \hat{j} \)
    c) \( \hat{k} \)
    d) \( 2\hat{k} \)
    Answer: c) \( \hat{k} \)

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