ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Vectors (Set 1): 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 3i − 4j is:
Answer: C. 5
Explanation: |3i − 4j| = √(9 + 16) = 5.
Q2. The unit vector in the direction of 2i − j + 2k is:
Answer: B. (2i − j + 2k)/3
Explanation: |2i − j + 2k| = √(4 + 1 + 4) = 3, so the unit vector is (2i − j + 2k)/3.
Q3. If a = i + 2j + 3k and b = 2i − j + k, then a·b equals:
Answer: B. 3
Explanation: a·b = (1)(2) + (2)(−1) + (3)(1) = 2 − 2 + 3 = 3.
Q4. The vector product j × i equals:
Answer: C. −k
Explanation: i × j = k and the cross product is anti-commutative, so j × i = −k.
Q5. If a = i − j and b = j − k, what angle does a make with b?
Answer: C. 120°
Explanation: (i − j)·(j − k) = 0 − 1 + 0 = −1 and each magnitude is √2, so cos θ = −1/2 and θ = 120°.
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Q6. For what value of λ are 2i + λj + k and i − 2j + 3k perpendicular?
Q7. The scalar projection of a = 2i + j + 3k on b = 2i − j + 2k is:
Q8. The area of the parallelogram with adjacent sides a = i + j and b = j + k is:
Q9. The area of the triangle with vertices O(0, 0, 0), A(1, 0, 0) and B(0, 2, 0) is:
Q10. The volume of the parallelepiped with edges a = i, b = 2j and c = i + j + 3k is:
Q11. For what value of λ are the vectors i + 2j + 3k, 2i + λj + 4k and i + j + k coplanar?
Q12. The direction cosines of the vector i + j + k are:
Q13. If a vector makes angles α, β and γ with the coordinate axes, then cos²α + cos²β + cos²γ equals:
Q14. The position vector of the midpoint of A(2, 4, 6) and B(4, 0, 2) is:
Q15. A force F = 2i + 3j + k moves a particle from (1, 1, 1) to (3, 2, 4). The work done is:
Q16. A force F = 3j acts at a point whose position vector relative to O is r = 2i. The torque about O is:
Q17. Which statement about the cross product is always true?
Q18. For any vectors a and b, the value of a·(a × b) is:
Q19. The vector product k × j equals:
Q20. If |a| = 3, |b| = 4 and a is perpendicular to b, then |a + b| equals:
Q21. If |a| = 2, |b| = 3 and a·b = 3, the angle between a and b is:
Q22. For what value of λ are the vectors 2i + 3j − k and 4i + λj − 2k parallel?
Q23. The distance between the points P(1, 2, 3) and Q(3, 4, 4) is:
Q24. If A = (1, −1, 2) and B = (3, 2, 0), the vector AB is:
Q25. A vector of magnitude 10 in the direction of 3i + 4j is:
Q26. If |a| = 2, |b| = 5 and a·b = 6, then |a × b| equals:
Q27. A unit vector perpendicular to both i + j and j + k is:
Q28. Which statement is correct?
Q29. For a triangle ABC, the vector sum AB + BC + CA equals:
Q30. A and B have position vectors i and 4i + 3j. The point dividing AB internally in the ratio 2 : 1 has position vector: