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ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Vectors (Set 2)

Take the ECAT Mathematics — Vectors (Set 2): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers 30 ECAT…

What this test covers

Sample questions with answers

The first 5 questions of this test, with the correct answer and explanation.

  1. Q1. The magnitude of the vector 2i − 3j + 6k is:

    • A. 49
    • B. 11
    • C. 5
    • D. 7 ✓

    Answer: D. 7

    Explanation: |a| = √(4 + 9 + 36) = √49 = 7.

  2. Q2. The unit vector in the direction of 6i − 2j + 3k is:

    • A. (6i − 2j + 3k)/7 ✓
    • B. (6i + 2j + 3k)/7
    • C. (6i − 2j + 3k)/√7
    • D. (6i − 2j + 3k)/49

    Answer: A. (6i − 2j + 3k)/7

    Explanation: |a| = √(36 + 4 + 9) = 7, so â = (6i − 2j + 3k)/7.

  3. Q3. If a = 3i − j + 2k and b = i + 4j − k, then a·b equals:

    • A. −3 ✓
    • B. 9
    • C. −5
    • D. 3

    Answer: A. −3

    Explanation: a·b = 3(1) + (−1)(4) + 2(−1) = 3 − 4 − 2 = −3.

  4. Q4. The vector product i × k equals:

    • A. k
    • B. j
    • C. 0
    • D. −j ✓

    Answer: D. −j

    Explanation: Cyclic order i→j→k gives k × i = j, so i × k = −j.

  5. Q5. The angle between the vectors √3 i + j and i is:

    • A. 45°
    • B. 30° ✓
    • C. 60°
    • D. 90°

    Answer: B. 30°

    Explanation: cos θ = (√3)/(2·1) = √3/2, so θ = 30°.

The other 25 questions

Take the timed test (free, no sign-up) to see your score and the answer and explanation for every question.

  1. Q6. For what value of λ are the vectors λi + 2j − k and 3i − j + 4k perpendicular?

    • A. −2
    • B. 6
    • C. 2/3
    • D. 2

  2. Q7. The scalar projection of a = i + 2j + 2k on b = 3i + 4k is:

    • A. 3
    • B. 11
    • C. 11/3
    • D. 11/5

  3. Q8. (i + 2j) × (3i + k) equals:

    • A. 2i − j + 6k
    • B. 2i − j − 6k
    • C. −2i + j + 6k
    • D. 2i + j − 6k

  4. Q9. The area of the triangle with vertices A(1, 0, 0), B(0, 1, 0) and C(0, 0, 1) is:

    • A. 3/2
    • B. 1/2
    • C. √3/2
    • D. √3

  5. Q10. The volume of the parallelepiped with edges i + j, j + k and k + i is:

    • A. 2
    • B. 3
    • C. 1
    • D. 0

  6. Q11. Three non-zero vectors a, b and c are coplanar if and only if:

    • A. a·(b × c) = 0
    • B. a × b = b × c
    • C. |a| = |b| = |c|
    • D. a·b = b·c

  7. Q12. The direction cosines of the vector 2i − j + 2k are:

    • A. 2, −1, 2
    • B. 2/9, −1/9, 2/9
    • C. 2/3, −1/3, 2/3
    • D. 2/√5, −1/√5, 2/√5

  8. Q13. A line makes angles of 60° and 45° with the x- and y-axes. The acute angle it makes with the z-axis is:

    • A. 30°
    • B. 60°
    • C. 90°
    • D. 45°

  9. Q14. The point dividing the join of A(1, 2, 3) and B(4, 5, 6) internally in the ratio 2 : 1 has coordinates:

    • A. (7, 8, 9)
    • B. (3, 4, 5)
    • C. (2, 3, 4)
    • D. (5/2, 7/2, 9/2)

  10. Q15. A constant force F = 3i + 2j − k moves a particle through the displacement d = i + 3j + 2k. The work done is:

    • A. 5
    • B. 7
    • C. 11
    • D. 9

  11. Q16. A force F = 2k acts at the point with position vector r = i + j. The magnitude of its moment about the origin is:

    • A. 2
    • B. 4
    • C. 2√2
    • D. √2

  12. Q17. If a and b are non-zero vectors with a × b = 0, then:

    • A. |a| = |b|
    • B. a·b = 0
    • C. a is parallel to b
    • D. a is perpendicular to b

  13. Q18. For any vectors a and b, (a + b)·(a − b) equals:

    • A. 0
    • B. |a|² + |b|²
    • C. |a − b|²
    • D. |a|² − |b|²

  14. Q19. If a and b are unit vectors inclined at 60°, then |a + b| equals:

    • A. 2
    • B. √2
    • C. √3
    • D. 1

  15. Q20. If |a| = 5, |b| = 3 and a·b = 6, then |a − b| equals:

    • A. √28
    • B. √22
    • C. √46
    • D. 2

  16. Q21. For what value of λ is λi − 6j + 3k parallel to 2i + 4j − 2k?

    • A. 4
    • B. 3
    • C. −4
    • D. −3

  17. Q22. The distance between the points (2, −1, 3) and (−1, 3, 3) is:

    • A. 7
    • B. 5
    • C. 25
    • D. √7

  18. Q23. If |a| = 3, |b| = 4 and |a × b| = 6, the acute angle between a and b is:

    • A. 30°
    • B. 60°
    • C. 45°
    • D. 90°

  19. Q24. Find a vector of length 1 that is normal to the plane containing a = i + j + k and b = i − j.

    • A. (2i − j − k)/√6
    • B. (i + j − 2k)/√6
    • C. (i − j)/√2
    • D. (i + j + 2k)/√6

  20. Q25. For any vectors a and b, which relation always holds?

    • A. |a × b| = a·b
    • B. a·b = −(b·a)
    • C. a × b = b × a
    • D. a × b = −(b × a)

  21. Q26. The value of i·(k × j) is:

    • A. −1
    • B. 1
    • C. i
    • D. 0

  22. Q27. If A = (2, −1, 4) and B = (5, 3, 4), then |AB| equals:

    • A. √41
    • B. 25
    • C. 5
    • D. 7

  23. Q28. A vector of magnitude 6 in the direction of 2i − j + 2k is:

    • A. 2i − j + 2k
    • B. 4i − 2j + 4k
    • C. 6i − 3j + 6k
    • D. 12i − 6j + 12k

  24. Q29. The angle that the vector i + j + √2 k makes with the z-axis is:

    • A. 30°
    • B. 60°
    • C. 90°
    • D. 45°

  25. Q30. In a parallelogram ABCD, the diagonal vector AC equals:

    • A. AB − AD
    • B. AD − AB
    • C. AB + AD
    • D. ½(AB + AD)