ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Vectors (Set 7): 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. For two vectors, |a × b| = 8 and a·b = 6. Then the product |a||b| equals:
Answer: D. 10
Explanation: Lagrange: |a|²|b|² = |a × b|² + (a·b)² = 64 + 36 = 100, so |a||b| = 10.
Q2. The tetrahedron with vertices O(0, 0, 0), A(1, 2, 3), B(2, −1, 1) and C(3, 1, −2) has volume:
Answer: A. 5
Explanation: Volume = (1/6)|[OA OB OC]|. The determinant with rows (1, 2, 3), (2, −1, 1), (3, 1, −2) is 30, so the volume is 30/6 = 5.
Q3. The vector projection of a = 3i − j + 2k on b = i + 2j + 2k is:
Answer: C. (5/9)(i + 2j + 2k)
Explanation: a·b = 3 − 2 + 4 = 5 and |b|² = 9. The vector projection is (a·b/|b|²) b = (5/9)(i + 2j + 2k). (5/3 is the scalar projection.)
Q4. The points (1, 1, 1), (2, 3, 4), (0, −1, 1) and (k, 0, 2) are coplanar. Then k equals:
Answer: B. 1/2
Explanation: The vectors from (1, 1, 1) are (1, 2, 3), (−1, −2, 0) and (k − 1, −1, 1). Their determinant is 6k − 3, which is zero only for k = 1/2.
Q5. For a = i + j, b = i − j + k and c = 2j + k, the vector (a × b) × c equals:
Answer: A. 3i − j + 2k
Explanation: a × b = (1, −1, −2). Then (a × b) × c = (−1·1 − (−2)·2, −2·0 − 1·1, 1·2 − (−1)·0) = (3, −1, 2).
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Q6. For a = i + j, b = i − j + k and c = 2j + k, the value of (a × b)·(a × c) is:
Q7. A vector of magnitude 6 is perpendicular to both i + j and j + k and makes an acute angle with the z-axis. It is:
Q8. The points P(1, −2, 3), Q(3, 0, −1) and R(5, 2, k) are collinear. Then k equals:
Q9. The point dividing the join of A(−1, 4, 2) and B(9, −1, 7) internally in the ratio 3 : 2 is:
Q10. In the parallelogram ABCD, A = (1, 2, −1), B = (3, 5, 2) and C = (7, 4, 3). The vertex D is:
Q11. The cosine of the angle between a main diagonal of a cube and a diagonal of one of its faces meeting it at a vertex (e.g. i + j + k and i + j) is:
Q12. The cosine of the acute angle between two main diagonals of a cube, such as i + j + k and −i + j + k, is:
Q13. If [a b c] = 7, then the scalar triple product [a − b, b − c, c − a] equals:
Q14. If [a b c] = 3, then the scalar triple product [2a + b, b + 3c, c − a] equals:
Q15. Which condition on non-zero vectors a and b implies that a and b are perpendicular?
Q16. The unit vectors a, b, c satisfy a + b + c = 0. Then a·b + b·c + c·a equals:
Q17. Two vectors have |a| = 2, |b| = 1 and the angle between them is 60°. The cosine of the angle between a + b and a − b is:
Q18. The area of the triangle whose vertices have position vectors i, 2j and 3k is:
Q19. The volume of the parallelepiped with edges i + 2j, j + 2k and 2i + k is:
Q20. The component of a = i + 2j + 3k perpendicular to b = i + j + k is:
Q21. Forces i + 2j + 3k and 2i − j + k act on a particle which is in equilibrium under a third force as well. The magnitude of the third force is:
Q22. In the triangle with vertices A(2, 3, 3), B(1, 2, 3) and C(1, 3, 4), the angle ABC is:
Q23. For any vectors a, b, c, the vector (a × b) × (a × c) equals:
Q24. For a = i, b = j and c = 2i + 3j + 6k, the scalar projection of the vector a × b on c is:
Q25. Adjacent sides of a parallelogram are 2a + b and a − 3b, where |a × b| = 4. The area of the parallelogram is:
Q26. A line makes angles α, β, γ with the axes, with cos α = 1/3 and cos β = 2/3. Then cos²γ equals:
Q27. A force of magnitude 10 N acts along the direction i + 2j + 2k and displaces a body by 3i + 3j (metres). The work done is:
Q28. A force F = i + 2j + 3k acts at the point P(1, −1, 2). The magnitude of its moment about the point Q(1, 0, 0) is:
Q29. Two vectors a and b have |a| = 1, |b| = 2 and the angle between them is 120°. Then |a − b| equals:
Q30. For a = 2i − j + k and b = i + 2j + 2k, the vector a + λb is perpendicular to b when λ equals: