ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Vectors (Set 6): 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. If a = i + 2j + 2k and b = 2i − j + 2k, the scalar projection of a on b is:
Answer: A. 4/3
Explanation: a·b = 2 − 2 + 4 = 4 and |b| = 3, so the scalar projection of a on b is 4/3.
Q2. The vectors i + j and j + k are adjacent sides of a parallelogram. The angle between them is:
Answer: B. 60°
Explanation: cos θ = (a·b)/(|a||b|) = 1/(√2·√2) = 1/2, so θ = 60°.
Q3. For u = 2i + 3j − k and w = i − 2j + 4k, the vector u × w is:
Answer: D. 10i − 9j − 7k
Explanation: u × w = i(3·4 − (−1)(−2)) − j(2·4 − (−1)(1)) + k(2(−2) − 3·1) = 10i − 9j − 7k.
Q4. The vectors a = 2i + λj + k and b = i − 2j + 3k are perpendicular. Then λ equals:
Answer: D. 5/2
Explanation: a·b = 2 − 2λ + 3 = 0 ⇒ λ = 5/2.
Q5. The vectors i + 2j + 3k, 2i + j + k and 3i + lj + 4k are coplanar. Then l equals:
Answer: C. 3
Explanation: Coplanar means the scalar triple product is zero: 1(4 − l) − 2(8 − 3) + 3(2l − 3) = 5l − 15 = 0 ⇒ l = 3.
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Q6. Which of the following is a unit vector perpendicular to both i + j and j + k?
Q7. If |a| = 2, |b| = 3 and the angle between them is 60°, then |a − b|² equals:
Q8. For unit vectors a and b, |a + b| = √3. Then the angle between a and b is:
Q9. The work done by the force F = 2i + j − k in moving a particle from A(1, 1, 1) to B(3, 0, 2) is:
Q10. A force F = i + 2j + 3k acts at the point (1, −1, 2). The magnitude of its moment about the origin is:
Q11. The position vectors of A and B are 2i + 3j and 4i − j. The point dividing AB internally in the ratio 1 : 3 has position vector:
Q12. The direction cosines of the line joining A(1, 2, 3) to B(3, −2, 7) are:
Q13. For any two vectors a and b, |a × b|² + (a·b)² equals:
Q14. For three non-coplanar vectors a, b, c, the scalar triple product a·(b × c):
Q15. The sum of the vectors 2i − j + 2k and 4i + 3j − 2k has the unit vector:
Q16. A vector of magnitude 6 is parallel to 2i − 2j + k and points the same way. It is:
Q17. If a·b = a·c with a ≠ 0, then:
Q18. In a cube, the cosine of the angle between a diagonal and an edge meeting it at a vertex is:
Q19. The vectors 3i − 2j + k and 6i − 4j + λk are parallel for λ equal to:
Q20. The area of the parallelogram whose diagonals are 3i + j − 2k and i − 3j + 4k is:
Q21. Three vectors satisfy a + b + c = 0 with |a| = 3, |b| = 5 and |c| = 7. The angle between a and b is:
Q22. For a = 2i + 3j + 6k and b = i + 2j + 2k, the magnitude of the component of a perpendicular to b is:
Q23. The vector (i + j) × (j × k) equals:
Q24. A rigid body rotates with angular velocity ω = 2k. The speed of the particle at r = 3i + 4j is:
Q25. If a = i + 2j − k and b = 2i + k, then |a × b| equals:
Q26. The volume of the tetrahedron with vertices O(0, 0, 0), A(2, 0, 0), B(0, 3, 0) and C(1, 1, 4) is:
Q27. The unit vectors perpendicular to both i + j + k and i − j are:
Q28. The acute angle between lines with direction ratios (1, −1, 2) and (2, 1, −1) is:
Q29. For non-zero vectors a and b, (a + b)·(a − b) = 0 if and only if:
Q30. The line through A(1, 2, 3) and B(3, 4, 7) meets the xy-plane at the point: