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ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Vectors (Set 3)

Take the ECAT Mathematics — Vectors (Set 3): 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 + 2j − k is:

    • A. 9
    • B. 5
    • C. 3 ✓
    • D. √7

    Answer: C. 3

    Explanation: |a| = √(4 + 4 + 1) = √9 = 3.

  2. Q2. If a = i + 2j + 3k and b = 3i − 2j + k, then a·b equals:

    • A. −2
    • B. 10
    • C. 2 ✓
    • D. 6

    Answer: C. 2

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

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

    • A. −3i + j + 2k
    • B. i − j + 2k
    • C. 3i − j − 2k ✓
    • D. 3i + j − 2k

    Answer: C. 3i − j − 2k

    Explanation: a × b = (1·2 − 1·(−1))i − (1·2 − 1·1)j + (1·(−1) − 1·1)k = 3i − j − 2k.

  4. Q4. The angle between a = 3i + 4j and b = 4i − 3j is:

    • A. 90° ✓
    • B. 45°
    • C. 180°
    • D. 60°

    Answer: A. 90°

    Explanation: a·b = 12 − 12 = 0, so the vectors are perpendicular.

  5. Q5. The cosine of the angle between a = 2i + j − 2k and b = 3i − 4k is:

    • A. 2/15
    • B. 14/15 ✓
    • C. −14/15
    • D. 14/5

    Answer: B. 14/15

    Explanation: a·b = 6 + 0 + 8 = 14, |a| = 3, |b| = 5, so cos θ = 14/15.

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. The vector projection of a = 2i + 3j on b = i + j is:

    • A. (5/2)i
    • B. 5(i + j)
    • C. (5/√2)(i + j)
    • D. (5/2)(i + j)

  2. Q7. The vectors i + j + k, 2i − j + 3k and 4i + j + λk are coplanar when λ equals:

    • A. 3
    • B. 5
    • C. 2/3
    • D. −5

  3. Q8. For a = 2i − j, b = j + 3k and c = i + k, the scalar triple product a·(b × c) equals:

    • A. −5
    • B. −1
    • C. 5
    • D. 1

  4. Q9. The diagonals of a parallelogram are 3i + j − 2k and i − 3j + 4k. Its area is:

    • A. 5√2
    • B. 5√6
    • C. 5√3
    • D. 10√3

  5. Q10. The vertices of a triangle are A(1, 1, 1), B(3, 1, 1) and C(1, 4, 5). Its area is:

    • A. 5
    • B. 10
    • C. 15/2
    • D. 7/2

  6. Q11. A vector makes angles with the x- and y-axes whose direction cosines are 1/2 and 1/2. The third direction cosine n satisfies:

    • A. n = ±1/2
    • B. n = 1/2
    • C. n = ±√2
    • D. n = ±1/√2

  7. Q12. A = i + 2j and B = 3i + j + k. The point dividing AB externally in the ratio 3 : 1 has position vector:

    • A. 4i + ½j + (3/2)k
    • B. 5i + j + 2k
    • C. 2i + (3/2)j + ½k
    • D. (5/2)i + (3/2)j + ½k

  8. Q13. A force F = 2i − j + k moves a particle from (1, 0, 2) to (3, 1, 5). The work done is:

    • A. 8
    • B. 6
    • C. 4
    • D. −6

  9. Q14. A force i + j + k acts at the point (1, −1, 0). Its moment about the origin, r × F, is:

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

  10. Q15. A unit vector perpendicular to both a = i + 2j + k and b = 2i + j − k is:

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

  11. Q16. If |a| = 3, |b| = 4 and the angle between a and b is 60°, then |a − b| equals:

    • A. √13
    • B. √37
    • C. 5
    • D. 1

  12. Q17. For non-zero vectors a and b, if |a + b| = |a − b| then a and b are:

    • A. of equal magnitude
    • B. perpendicular
    • C. parallel
    • D. equal

  13. Q18. If |a| = 3, |b| = 5 and a·b = 9, then |a × b| equals:

    • A. √144 + 9
    • B. 6
    • C. 15
    • D. 12

  14. Q19. The points A(1, 2, 3), B(3, 5, 7) and C(7, 11, k) are collinear. Then k equals:

    • A. 15
    • B. 19
    • C. 13
    • D. 11

  15. Q20. In the triangle with vertices A(1, 1, 0), B(0, 0, 0) and C(1, 0, 1), the angle at B is:

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

  16. Q21. The scalar component of a = 2i + j − k along b = i + 2j + 2k is:

    • A. 2/3
    • B. 2
    • C. −2/3
    • D. 2/9

  17. Q22. Simplify (a + b) × (a − b), where × denotes the vector product:

    • A. a × a − b × b
    • B. −2(a × b)
    • C. 0
    • D. 2(a × b)

  18. Q23. The direction cosines of the line joining A(1, −2, 3) and B(4, 2, −9) are:

    • A. 3/√34, 4/√34, −12/√34
    • B. 3/13, 4/13, −12/13
    • C. 3, 4, −12
    • D. 1/13, 2/13, 3/13

  19. Q24. The angle between a diagonal of a cube and one of its edges meeting at the same vertex is:

    • A. cos⁻¹(1/3)
    • B. cos⁻¹(1/√3)
    • C. 45°
    • D. cos⁻¹(1/√2)

  20. Q25. A parallelogram ABCD has A(1, 2, 0), B(3, 3, 1) and C(4, 5, 3). The vertex D is:

    • A. (2, 4, 4)
    • B. (0, 0, −2)
    • C. (2, 4, 2)
    • D. (6, 6, 4)

  21. Q26. The centroid of the triangle with vertices (1, 1, 0), (3, 0, 3) and (2, 5, 3) is:

    • A. (2, 2, 2)
    • B. (2, 3, 1)
    • C. (3, 2, 2)
    • D. (6, 6, 6)

  22. Q27. A vector of magnitude 15 in the direction of 4i − 3j is:

    • A. 20i − 15j
    • B. 3i − 4j
    • C. 12i − 9j
    • D. 4i − 3j

  23. Q28. The value of (i × j) × j is:

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

  24. Q29. Which of the following pairs of vectors is perpendicular?

    • A. 2i + k and i + j + k
    • B. i + j + k and i + 2j + 3k
    • C. i − j + 2k and 2i + 2j + k
    • D. i + 2j + 3k and 2i − j

  25. Q30. If a, b are vectors with |a| = |b| = |a + b|, the angle between a and b is:

    • A. 90°
    • B. 60°
    • C. 120°
    • D. 30°