JEE (Main) Mathematics - Vector Algebra MCQs

16. What is the direction cosines of the vector \( \mathbf{A} = 3\hat{i} + 4\hat{j} + 12\hat{k} \)?
    • a) \( \frac{3}{\sqrt{9+16+144}} , \frac{4}{\sqrt{9+16+144}} , \frac{12}{\sqrt{9+16+144}} \)
    • b) \( \frac{3}{\sqrt{9+16}} , \frac{4}{\sqrt{9+16}} , \frac{12}{\sqrt{9+16}} \)
    • c) \( \frac{3}{\sqrt{9+16+49}} , \frac{4}{\sqrt{9+16+49}} , \frac{12}{\sqrt{9+16+49}} \)
    • d) \( \frac{3}{\sqrt{9+49}} , \frac{4}{\sqrt{9+49}} , \frac{12}{\sqrt{9+49}} \)

    Answer: a) \( \frac{3}{\sqrt{9+16+144}} , \frac{4}{\sqrt{9+16+144}} , \frac{12}{\sqrt{9+16+144}} \)

17. What is the sum of vectors \( \mathbf{A} = 2\hat{i} + \hat{j} \) and \( \mathbf{B} = 3\hat{i} + 4\hat{j} \)?
    • a) \( 5\hat{i} + 5\hat{j} \)
    • b) \( 5\hat{i} + 3\hat{j} \)
    • c) \( 5\hat{i} + 4\hat{j} \)
    • d) \( 5\hat{i} + 6\hat{j} \)

    Answer: a) \( 5\hat{i} + 5\hat{j} \)

18. What is the direction ratio of the vector \( \mathbf{A} = 4\hat{i} - 2\hat{j} + 5\hat{k} \)?
    • a) \( 4, -2, 5 \)
    • b) \( 2, 4, 5 \)
    • c) \( -4, 2, -5 \)
    • d) \( 4, 2, 5 \)

    Answer: a) \( 4, -2, 5 \)

19. If the magnitude of a vector is \( 10 \) and the direction cosines are \( \frac{3}{5}, \frac{4}{5}, 0 \), what is the vector?
    • a) \( 6\hat{i} + 8\hat{j} \)
    • b) \( 3\hat{i} + 4\hat{j} \)
    • c) \( 10\hat{i} + 8\hat{j} \)
    • d) \( 8\hat{i} + 6\hat{j} \)

    Answer: a) \( 6\hat{i} + 8\hat{j} \)

20. The scalar product (dot product) of two vectors \( \mathbf{A} = 2\hat{i} + 3\hat{j} \) and \( \mathbf{B} = 4\hat{i} - \hat{j} \) is:
    • a) 11
    • b) 7
    • c) 6
    • d) 1

    Answer: b) 7

21. What is the angle between the vectors \( \mathbf{A} = \hat{i} + \hat{j} \) and \( \mathbf{B} = 3\hat{i} - 4\hat{j} \)?
    • a) 45°
    • b) 90°
    • c) 60°
    • d) 30°

    Answer: b) 90°

22. If vector \( \mathbf{A} = 3\hat{i} + 2\hat{j} - 4\hat{k} \) and vector \( \mathbf{B} = 2\hat{i} + 3\hat{j} + \hat{k} \), what is their cross product \( \mathbf{A} \times \mathbf{B} \)?
    • a) \( 10\hat{i} - 10\hat{j} + 7\hat{k} \)
    • b) \( 5\hat{i} - 4\hat{j} + 6\hat{k} \)
    • c) \( -10\hat{i} + 10\hat{j} + 7\hat{k} \)
    • d) \( -10\hat{i} - 10\hat{j} + 7\hat{k} \)

    Answer: a) \( 10\hat{i} - 10\hat{j} + 7\hat{k} \)

23. The projection of a vector \( \mathbf{A} = 5\hat{i} + 3\hat{j} \) on the direction of vector \( \mathbf{B} = \hat{i} + 2\hat{j} \) is:
    • a) \( 4 \)
    • b) \( 5 \)
    • c) \( 3 \)
    • d) \( 7 \)

    Answer: a) 4

24. Which of the following is true for two perpendicular vectors \( \mathbf{A} \) and \( \mathbf{B} \)?
    • a) \( \mathbf{A} \cdot \mathbf{B} = 1 \)
    • b) \( \mathbf{A} \cdot \mathbf{B} = 0 \)
    • c) \( \mathbf{A} \times \mathbf{B} = 0 \)
    • d) \( \mathbf{A} \times \mathbf{B} = 1 \)

    Answer: b) \( \mathbf{A} \cdot \mathbf{B} = 0 \)

25. The sum of vectors \( \mathbf{A} = 2\hat{i} + 3\hat{j} + 4\hat{k} \) and \( \mathbf{B} = 5\hat{i} - 2\hat{j} + \hat{k} \) results in a vector of components:
    • a) \( 7\hat{i} + \hat{j} + 5\hat{k} \)
    • b) \( 7\hat{i} + 5\hat{j} + 5\hat{k} \)
    • c) \( 3\hat{i} + \hat{j} + 4\hat{k} \)
    • d) \( 7\hat{i} - \hat{j} + 3\hat{k} \)

    Answer: a) \( 7\hat{i} + \hat{j} + 5\hat{k} \)

26. The cross product \( \mathbf{A} \times \mathbf{B} \) of vectors \( \mathbf{A} = \hat{i} + 2\hat{j} \) and \( \mathbf{B} = 2\hat{i} - \hat{j} \) is:
    • a) \( 3\hat{k} \)
    • b) \( -3\hat{k} \)
    • c) \( 4\hat{k} \)
    • d) \( -4\hat{k} \)

    Answer: a) \( 3\hat{k} \)

27. The angle between two vectors \( \mathbf{A} = \hat{i} + \hat{j} \) and \( \mathbf{B} = -\hat{i} + 2\hat{j} \) is:
    • a) 90°
    • b) 45°
    • c) 60°
    • d) 120°

    Answer: a) 90°

28. The scalar product of two vectors \( \mathbf{A} = 2\hat{i} + 3\hat{j} \) and \( \mathbf{B} = 3\hat{i} - 4\hat{j} \) is:
    • a) 0
    • b) 5
    • c) 6
    • d) -5

    Answer: a) 0

29. If two vectors \( \mathbf{A} = \hat{i} + 2\hat{j} + 3\hat{k} \) and \( \mathbf{B} = 2\hat{i} + 3\hat{j} + \hat{k} \) are parallel, then their cross product is:
    • a) \( 0 \)
    • b) \( 2\hat{i} + \hat{j} + 3\hat{k} \)
    • c) \( 1\hat{i} + 4\hat{j} \)
    • d) \( \hat{i} + 3\hat{j} + 4\hat{k} \)

    Answer: a) \( 0 \)

30. The projection of vector \( \mathbf{A} = 5\hat{i} + 2\hat{j} \) on vector \( \mathbf{B} = 3\hat{i} + 4\hat{j} \) is:
    • a) \( \frac{15}{5} \)
    • b) \( \frac{7}{4} \)
    • c) \( \frac{5}{3} \)
    • d) \( \frac{10}{5} \)

    Answer: b) \( \frac{7}{4} \)

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