Take timed practice tests on Vectors for JEE Main and JEE Advanced with session-wise drills, score review, and explanation-led revision.
Six 20-question timed sessions plus one 60-question module test. Each question is original and calibrated from the uploaded material pattern without copying PDF wording.
Ready
1. The magnitude of vector a⃗ = 3î + 4ĵ is:
Explanation: |a⃗| = √(3² + 4²) = √(9 + 16) = √25 = 5.
2. The unit vector in the direction of a⃗ = 2î + 2ĵ + k̂ is:
Explanation: |a⃗| = √(4 + 4 + 1) = √9 = 3. Unit vector = a⃗/|a⃗| = (2î + 2ĵ + k̂)/3.
3. If a⃗ = î + 2ĵ and b⃗ = 3î − ĵ, then a⃗ · b⃗ is:
Explanation: a⃗ · b⃗ = (1)(3) + (2)(−1) + (0)(0) = 3 − 2 = 1.
4. The position vector of the midpoint of A(2, 4, 6) and B(4, 8, 10) is:
Explanation: Midpoint = ((2+4)/2)î + ((4+8)/2)ĵ + ((6+10)/2)k̂ = 3î + 6ĵ + 8k̂.
5. Vectors a⃗ = 2î + 4ĵ and b⃗ = î + 2ĵ are:
Explanation: b⃗ = (1/2)a⃗. When one vector is a scalar multiple of another, they are parallel (or anti-parallel if the scalar is negative). Here the scalar 1/2 > 0, so they are parallel.
6. If a⃗ = î and b⃗ = ĵ, then a⃗ × b⃗ is:
Explanation: Using the right-hand rule and the cyclic property: î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ.
7. The angle between vectors a⃗ = î + ĵ and b⃗ = î − ĵ is:
Explanation: a⃗ · b⃗ = (1)(1) + (1)(−1) = 0. Since dot product = 0, the vectors are perpendicular (90°).
8. If [a⃗, b⃗, c⃗] = 0 (scalar triple product), then a⃗, b⃗, c⃗ are:
Explanation: The scalar triple product a⃗ · (b⃗ × c⃗) represents the volume of the parallelepiped. If it equals 0, the volume is 0, meaning the three vectors are coplanar (lie in the same plane).
9. The area of parallelogram with sides a⃗ and b⃗ is:
Explanation: The area of a parallelogram with sides a⃗ and b⃗ is |a⃗ × b⃗| = |a⃗||b⃗|sin θ, where θ is the angle between them.
10. If P divides AB in ratio m:n internally, the position vector of P is:
Explanation: Internal section formula: P = (m·b⃗ + n·a⃗)/(m+n) where a⃗ = position of A, b⃗ = position of B, and ratio AP:PB = m:n.
11. The projection of a⃗ on b⃗ is:
Explanation: Projection of a⃗ onto b⃗ = |a⃗|cos θ = |a⃗|(a⃗·b⃗)/(|a⃗||b⃗|) = (a⃗·b⃗)/|b⃗|.
12. If a⃗ + b⃗ + c⃗ = 0 and |a⃗| = 3, |b⃗| = 5, |c⃗| = 7, then the angle between a⃗ and b⃗ is:
Explanation: a⃗ + b⃗ = −c⃗ → |a⃗+b⃗|² = |c⃗|² = 49. |a⃗|² + 2a⃗·b⃗ + |b⃗|² = 49 → 9 + 2a⃗·b⃗ + 25 = 49 → a⃗·b⃗ = 15/2. cos θ = (a⃗·b⃗)/(|a⃗||b⃗|) = (15/2)/(15) = 1/2 → θ = 60°.
13. Three vectors are linearly independent if:
Explanation: Three vectors a⃗, b⃗, c⃗ are linearly independent iff [a⃗, b⃗, c⃗] ≠ 0 (non-zero scalar triple product). If [a⃗, b⃗, c⃗] = 0, they are coplanar (linearly dependent).
14. If a⃗ × b⃗ = c⃗ and b⃗ × c⃗ = a⃗, then:
Explanation: From a⃗ × b⃗ = c⃗: c⃗ ⊥ a⃗ and c⃗ ⊥ b⃗. From b⃗ × c⃗ = a⃗: a⃗ ⊥ b⃗ and a⃗ ⊥ c⃗. So all three are mutually perpendicular. Also |c⃗| = |a⃗||b⃗|sin(90°) = |a⃗||b⃗|. Similarly |a⃗| = |b⃗||c⃗|. This gives |a⃗| = |b⃗||c⃗| = |b⃗|·|a⃗||b⃗| = |a⃗||b⃗|² → |b⃗|² = 1 → |b⃗| = 1. Similarly |a⃗| = |c⃗| = 1.
15. The vector equation of a line through point A (position vector a⃗) parallel to vector b⃗ is:
Explanation: Any point on the line through A parallel to b⃗ has position vector r⃗ = a⃗ + λb⃗ for some scalar λ ∈ ℝ. This is the parametric vector equation of the line.
16. The dot product of perpendicular vectors is:
Explanation: a⃗ · b⃗ = |a⃗||b⃗|cos θ. If θ = 90°, cos 90° = 0, so a⃗ · b⃗ = 0.
17. If a⃗ = 2î − ĵ + 2k̂, then |a⃗| is:
Explanation: |a⃗| = √(4 + 1 + 4) = √9 = 3.
18. The vector perpendicular to both î + ĵ and ĵ + k̂ is:
Explanation: Cross product (î+ĵ) × (ĵ+k̂) = |î ĵ k̂; 1 1 0; 0 1 1| = î(1−0) − ĵ(1−0) + k̂(1−0) = î − ĵ + k̂.