JEE (Main) Mathematics - Scalar and Vector Products MCQs

31. What is the scalar (dot) product of two vectors \( \mathbf{A} = 2\hat{i} + 3\hat{j} \) and \( \mathbf{B} = 4\hat{i} + 5\hat{j} \)?
    • a) 23
    • b) 22
    • c) 20
    • d) 21
    Answer: a) 23
32. The cross product of two vectors \( \mathbf{A} = 2\hat{i} + 3\hat{j} + \hat{k} \) and \( \mathbf{B} = 4\hat{i} - \hat{j} + 2\hat{k} \) is:
    • a) \( 7\hat{i} - 10\hat{j} - 10\hat{k} \)
    • b) \( 10\hat{i} - 10\hat{j} - 7\hat{k} \)
    • c) \( 10\hat{i} + 10\hat{j} - 7\hat{k} \)
    • d) \( 10\hat{i} + 7\hat{j} + 10\hat{k} \)
    Answer: b) \( 10\hat{i} - 10\hat{j} - 7\hat{k} \)
33. The dot product of two vectors is zero. This means that the vectors are:
    • a) Parallel
    • b) Orthogonal
    • c) Perpendicular
    • d) Collinear
    Answer: b) Orthogonal
34. What is the magnitude of the cross product \( \mathbf{A} \times \mathbf{B} \) if \( \mathbf{A} = 3\hat{i} + 4\hat{j} \) and \( \mathbf{B} = 2\hat{i} - \hat{j} \)?
    • a) 6
    • b) 12
    • c) 10
    • d) 4
    Answer: a) 6
35. The scalar product of two vectors \( \mathbf{A} = \hat{i} + \hat{j} + \hat{k} \) and \( \mathbf{B} = 2\hat{i} + 3\hat{j} + 4\hat{k} \) is:
    • a) 6
    • b) 7
    • c) 10
    • d) 9
    Answer: b) 7
36. Which of the following represents the direction of the cross product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \)?
    • a) Parallel to \( \mathbf{A} \)
    • b) Parallel to \( \mathbf{B} \)
    • c) Perpendicular to the plane containing \( \mathbf{A} \) and \( \mathbf{B} \)
    • d) Perpendicular to \( \mathbf{A} \) and \( \mathbf{B} \)
    Answer: c) Perpendicular to the plane containing \( \mathbf{A} \) and \( \mathbf{B} \)
37. If \( \mathbf{A} \cdot \mathbf{B} = 0 \), then the vectors \( \mathbf{A} \) and \( \mathbf{B} \) are:
    • a) Parallel
    • b) Perpendicular
    • c) Collinear
    • d) None of the above
    Answer: b) Perpendicular
38. The vector product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is:
    • a) Commutative
    • b) Associative
    • c) Distributive
    • d) None of the above
    Answer: c) Distributive
39. If the angle between two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is \( 90^\circ \), what is the value of \( \mathbf{A} \cdot \mathbf{B} \)?
    • a) 1
    • b) 0
    • c) -1
    • d) 10
    Answer: b) 0
40. The scalar product of two vectors \( \mathbf{A} = \hat{i} + 2\hat{j} \) and \( \mathbf{B} = 3\hat{i} + 4\hat{j} \) is:
    • a) 11
    • b) 10
    • c) 9
    • d) 12
    Answer: a) 11
41. The magnitude of the cross product of two vectors is:
    • a) Zero when the vectors are parallel
    • b) Zero when the vectors are perpendicular
    • c) Maximum when the vectors are perpendicular
    • d) None of the above
    Answer: a) Zero when the vectors are parallel
42. If \( \mathbf{A} = 2\hat{i} + 3\hat{j} + 4\hat{k} \) and \( \mathbf{B} = \hat{i} + 2\hat{j} + 3\hat{k} \), then \( \mathbf{A} \times \mathbf{B} \) is:
    • a) \( -3\hat{i} + 6\hat{j} - \hat{k} \)
    • b) \( 5\hat{i} - 2\hat{j} + 6\hat{k} \)
    • c) \( 2\hat{i} + 4\hat{j} - \hat{k} \)
    • d) \( 4\hat{i} - 5\hat{j} + 7\hat{k} \)
    Answer: a) \( -3\hat{i} + 6\hat{j} - \hat{k} \)
43. The dot product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is defined as:
    • a) \( \mathbf{A} \times \mathbf{B} \)
    • b) \( |\mathbf{A}||\mathbf{B}| \cos \theta \)
    • c) \( |\mathbf{A}| |\mathbf{B}| \sin \theta \)
    • d) \( |\mathbf{A}| |\mathbf{B}| \)
    Answer: b) \( |\mathbf{A}||\mathbf{B}| \cos \theta \)
44. Which of the following is a property of the vector product (cross product)?
    • a) It is commutative
    • b) It is associative
    • c) It is distributive
    • d) None of the above
    Answer: c) It is distributive
45. If \( \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta \), then \( \theta \) is:
    • a) The angle between the two vectors
    • b) The sum of the vectors
    • c) The difference of the vectors
    • d) The cross product of the vectors
    Answer: a) The angle between the two vectors

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