ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Vectors (Set 4): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers 30 ECAT…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. The magnitude of the vector i + 4j + 8k is
Answer: B. 9
Explanation: |v| = √(1 + 16 + 64) = √81 = 9.
Q2. If a = 2i − j + 3k and b = i + 4j − 2k, then a·b equals
Answer: C. −8
Explanation: a·b = (2)(1) + (−1)(4) + (3)(−2) = 2 − 4 − 6 = −8.
Q3. The unit vector in the direction of 2i − 3j + 6k is
Answer: D. (2i − 3j + 6k)/7
Explanation: |v| = √(4 + 9 + 36) = 7, so the unit vector is v/7.
Q4. i × k equals
Answer: C. −j
Explanation: Using i × j = k, j × k = i, k × i = j, we get i × k = −(k × i) = −j.
Q5. The midpoint of P(−2, 5, 1) and Q(6, −1, 7) has position vector
Answer: D. 2i + 2j + 4k
Explanation: The midpoint is half the sum: ((−2 + 6)/2, (5 − 1)/2, (1 + 7)/2) = (2, 2, 4).
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Q6. The cosine of the angle between 2i + j − 2k and 3j + 4k is
Q7. The scalar projection of 2i + 3j + 6k on i + 2j + 2k is
Q8. The area of the parallelogram with adjacent sides a = i + 2j + 3k and b = 2i − j + k is
Q9. The area of the triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) is
Q10. The volume of the parallelepiped with edges a = i + j, b = j + k, c = i + k is
Q11. A line makes angles of 60° and 45° with the x-axis and y-axis. An acute angle it can make with the z-axis is
Q12. A force F = 2i − j + 3k moves a particle from A(1, 1, 1) to B(4, 3, 2). The work done is
Q13. A force F = i + 2j + 3k acts at the point (1, 0, 2). Its torque about the origin is
Q14. The vectors λi + 2j − k and 3i + λj + 4k are perpendicular when λ equals
Q15. The vectors 2i − 4j + 6k and −i + mj − 3k are parallel when m equals
Q16. The vectors i + 2j + 3k, 2i + j + λk and i − j + 2k are coplanar when λ equals
Q17. If a = i + j and b = j + k, then a × b equals
Q18. If |a| = 3, |b| = 4 and a·b = 6, then |a × b| equals
Q19. If |a| = |b| = |a + b| = 1, the angle between a and b is
Q20. If |a| = 5, |b| = 12 and |a + b| = |a − b|, then |a + b| equals
Q21. The vector projection of a = 3i + 4j on b = i + j is
Q22. If a = i, b = j and c = i + j, then a × (b × c) equals
Q23. A force F = 3i − j + 2k acts at the point (1, 2, 3). The magnitude of its moment about the point (0, 1, 1) is
Q24. The point dividing A(1, 2, 3) and B(7, 5, −3) internally in the ratio 2 : 1 has position vector
Q25. A parallelogram has adjacent sides 3i and 4j. The cosine of the acute angle between its diagonals is
Q26. The perpendicular distance of the point (1, 2, 3) from the line through the origin in the direction i + j + k is
Q27. If a = 2i + j − k and b = i − j + 2k, then (a + b) × (a − b) equals
Q28. The area of the triangle with vertices A(1, 1, 1), B(2, 3, 5) and C(4, 2, 3) is
Q29. The cosine of the angle between a diagonal of a cube and a diagonal of one of its faces (meeting at a vertex) is
Q30. The volume of the tetrahedron with concurrent edges a = i, b = i + j and c = i + j + k is