20 NEET‑level vector MCQs on dot and cross products with instant feedback and detailed solutions.

20 NEET‑Level Vector Numericals: Dot & Cross Product Quiz

1. If (a = 2i + 3j – k) and (b = i – 4j + 2k), find (a·b).

–12 –2 0 5

Detailed Solution:
1) Write components: a = (2,3,–1), b = (1,–4,2).
2) Multiply: 2×1 = 2; 3×(–4)=–12; (–1)×2=–2.
3) Sum: 2 –12 –2 = –12.
So (a·b) = –12.

2. If |u| = 3, |v| = 4 and u ⟂ v, find |u × v|.

7 12 5 1

Detailed Solution:
|u×v| = |u||v| sin90° = 3×4×1 = 12.

3. Compute (a·(b×c)) for a=(i+2j), b=(3j+k), c=(2k–i).

0 3 4 5

Detailed Solution:
b×c = det|i j k;0 3 1;-1 0 2| =6i–j+3k;
a·(...) =1×6+2×(–1)+0=4.

4. Find the angle between a=(1,1,1) and b=(1,2,3).

25° 22° 30° 20°

Detailed Solution:
a·b=6; |a|=√3; |b|=√14;
cosθ=6/√42≈0.926⇒θ≈22°.

5. Scalar projection of a=(3,4,0) on b=(4,0,3).

3 2.4 4 1.8

Detailed Solution:
a·b=12; |b|=5; projection=12/5=2.4.

6. Compute i × j.

-k k i j

Detailed Solution:
By right‑hand rule, i×j=k.

7. Area of parallelogram by a=(2,0,1), b=(1,2,0).

√21 √14 5 6

Detailed Solution:
a×b=(-2,1,4); |...|=√21.

8. Are a=(2,-3,1) and b=(3,2,1) perpendicular?

No Yes Cannot say Only if scaled

Detailed Solution:
a·b=1⇒ not perpendicular.

9. Compute a×b for a=(1,2,3), b=(4,5,6).

(3,-6,3) (-3,6,-3) (3,6,3) (-3,-6,3)

Detailed Solution:
a×b=(-3,6,-3).

10. (a×b)·c for a=(1,0,0), b=(0,1,0), c=(1,1,1).

0 2 3 1

Detailed Solution:
a×b=k; k·c=1.

11. If |a| = 2, |b| = 3 and angle between them is 60°, find a·b.

2 3 6 √6

Detailed Solution:
1) Formula: a·b = |a||b| cosθ.
2) Substitute: = 2 × 3 × cos60° = 6 × 0.5 = 3.

12. Given a·b = 6, |a| = 2, |b| = 3, find the angle between a and b.

90° 0° 30° 60°

Detailed Solution:
1) cosθ = (a·b)/(|a||b|) = 6/(2×3) = 1.
2) θ = arccos(1) = 0°.

13. Find k such that a = (k,2,3) is perpendicular to b = (1,0,-1).

-3 3 2 -2

Detailed Solution:
1) Perpendicular ⇒ dot = 0: k×1 + 2×0 + 3×(-1) = 0.
2) Simplify: k - 3 = 0 ⇒ k = 3.

14. If a=(1,2,3) and b=(2,4,6), find |a×b|.

√14 0 6 √54

Detailed Solution:
1) b = 2a ⇒ vectors parallel ⇒ cross = 0.
2) |a×b| = 0.

15. Volume of parallelepiped formed by a=(1,0,0), b=(0,2,0), c=(0,0,3).

1 6 0 2

Detailed Solution:
1) Compute b×c = (2×3 in correct orientation) = ??? Actually b×c=(6,0,0).
2) a·(b×c)=1×6+0+0=6 ⇒ volume = 6.

16. Find angle between a=(1,-1,1) and b=(1,1,-1).

60° 90° 109.47° 120°

Detailed Solution:
1) a·b=1-1-1=-1;
2) |a|=|b|=√3;
3) cosθ=-1/3 ⇒ θ≈109.47°.

17. Are a=(1,1,1), b=(2,3,4), c=(3,4,5) coplanar?

Yes, if a·(b×c)=0 Only if |b×c|=0 If b·(a×c)=1 If a+b+c=0

Detailed Solution:
1) b×c = (-1,2,-1);
2) a·(b×c)=1*(-1)+1*2+1*(-1)=0 ⇒ coplanar.

18. Find |a×b| for a=(1,2,0), b=(2,0,2).

4 6 √8 √12

Detailed Solution:
1) a×b=(4,-2,-4);
2) magnitude = √(16+4+16)=√36=6.

19. Unit vector perpendicular to a=(1,2,3) and b=(4,5,6).

(-3,6,-3)/3 (-1,2,-1)/√6 (3,-6,3)/√54 (1,-2,1)/6

Detailed Solution:
1) a×b=(-3,6,-3);
2) |a×b|=√(9+36+9)=√54=3√6;
3) unit = (-3,6,-3)/(3√6)=(-1,2,-1)/√6.

20. Area of triangle formed by a=(2,3,1) and b=(1,-1,2).

(√42)/2 √83 (√21)/2 (√83)/2

Detailed Solution:
1) a×b=(7,-3,-5);
2) |a×b|=√(49+9+25)=√83;
3) Area = ½√83.