JEE Main Vectors and 3D Geometry 2027 — Dot Product, Cross Product, Lines, Planes and 40 Practice MCQs

Last Updated: May 2026

JEE 2027 — Vectors and 3D Geometry overview

For JEE Main 2027 Vectors and 3D Geometry, expect 4-5 questions in total — 2-3 from vector algebra/products and 2 from 3D coordinate geometry (lines, planes, distance). The chapter is heavy on formula-recall but light on derivation; mastering 12 formulas can secure 95%+ accuracy.

Vector Algebra Quick Reference

Essential Vector Identities

• Coplanarity test: [a b c] = 0
• Perpendicularity: a·b = 0
• Parallel vectors: a×b = 0
• Projection of a on b: (a·b)/|b|
• Component of a along b: ((a·b)/|b|²)b
• Angle between vectors: cosθ = (a·b)/(|a||b|)
• Area of triangle (vertices A, B, C): ½|AB×AC|

3D Geometry — Lines

A line passing through point P(x₀, y₀, z₀) with direction ratios (a, b, c):

• Vector form: r = a + λb
• Cartesian form: (x – x₀)/a = (y – y₀)/b = (z – z₀)/c
• Two-point form: Line through (x₁,y₁,z₁) and (x₂,y₂,z₂): direction = (x₂–x₁, y₂–y₁, z₂–z₁)

3D Geometry — Planes

Distance Formulas

• Point to plane: |ax₀ + by₀ + cz₀ + d| / √(a²+b²+c²)
• Point to line: |AP×b| / |b| (where A is a point on the line, P is the external point, b is direction)
• Distance between parallel lines: |(a₂−a₁)×b| / |b|
• Distance between skew lines: |[a₂−a₁, b₁, b₂]| / |b₁×b₂|
• Angle between two planes: cosθ = (a₁a₂+b₁b₂+c₁c₂)/(√(a₁²+b₁²+c₁²)·√(a₂²+b₂²+c₂²))
• Angle between line and plane: sinθ = (al+bm+cn)/(|n|·|b|)

40 Practice MCQs

1. If a = i+2j+3k, |a| = (A) √14 (B) 14 (C) √13 (D) 6
2. Unit vector along i+j+k = (A) (i+j+k)/3 (B) (i+j+k)/√3 (C) (i+j+k) (D) (i-j+k)/√3
3. Dot product of i and j = (A) 1 (B) 0 (C) –1 (D) k
4. i × j = (A) –k (B) k (C) 0 (D) 1
5. If a·b = 0, vectors are — (A) parallel (B) perpendicular (C) collinear (D) coincident
6. Vectors (1,2,3) and (4,5,6): a·b = (A) 32 (B) 14 (C) 21 (D) 26
7. Cross product magnitude (1,0,0)×(0,1,0) = (A) 0 (B) 1 (C) 2 (D) k
8. Scalar triple [i j k] = (A) 0 (B) 1 (C) –1 (D) 3
9. If [a b c] = 0, vectors are — (A) coplanar (B) perpendicular (C) parallel (D) random
10. Area of triangle with sides a, b: = (A) ab (B) ½|a×b| (C) a·b (D) |a||b|
11. Angle between (1,1,0) and (1,0,1) = (A) 30° (B) 60° (C) 90° (D) 45°
12. Direction cosines satisfy — (A) l+m+n=1 (B) l²+m²+n²=1 (C) lmn=1 (D) l-m-n=0
13. Plane x+y+z=1 has normal — (A) (1,1,1) (B) (0,0,0) (C) (1,0,0) (D) (1,1,0)
14. Distance from origin to plane x+y+z=√3 = (A) 1 (B) √3 (C) 1/√3 (D) 3
15. Line (x-1)/2 = (y-2)/3 = (z-3)/4 has direction — (A) (1,2,3) (B) (2,3,4) (C) (1,1,1) (D) (3,4,5)
16. Volume of parallelepiped with edges (1,0,0),(0,1,0),(0,0,1) = (A) 0 (B) 1 (C) 3 (D) 6
17. Two skew lines lie in — (A) same plane (B) different planes (C) parallel planes (D) cannot say
18. Angle between line direction (1,1,1) and plane x+y+z=1 = (A) 0° (B) 30° (C) 60° (D) 90°
19. Foot of perpendicular from (1,1,1) to plane x+y+z=0 = (A) (0,0,0) (B) (–1,–1,–1) (C) (1,1,1) (D) origin
20. Vector triple a×(b×c) where a=b=c is — (A) 0 (B) a²a (C) 1 (D) c
21. If a×b = 0 and a≠0, b≠0, then — (A) a∥b (B) a⊥b (C) a=b (D) a+b=0
22. The angle between (i+j) and (i-j) is — (A) 0° (B) 45° (C) 90° (D) 180°
23. If a·b = |a||b|, the angle is — (A) 0° (B) 90° (C) 180° (D) 60°
24. If a·b = –|a||b|, the angle is — (A) 0° (B) 90° (C) 180° (D) 270°
25. Area of parallelogram with diagonals d₁, d₂ = (A) d₁·d₂ (B) ½|d₁×d₂| (C) |d₁×d₂| (D) d₁d₂
26. Two planes are perpendicular if — (A) n₁·n₂=0 (B) n₁×n₂=0 (C) n₁=n₂ (D) n₁=–n₂
27. The cosine of angle between two lines (a₁,b₁,c₁) and (a₂,b₂,c₂) — (A) Σa₁a₂ (B) Σa₁a₂/(|d₁||d₂|) (C) a₁+b₁+c₁ (D) 0
28. Number of direction ratios needed to specify a line — (A) 1 (B) 2 (C) 3 (D) 4
29. Position vector of midpoint of AB where A(1,2,3), B(3,4,5) = (A) (2,3,4) (B) (1,2,3) (C) (4,6,8) (D) (0,0,0)
30. Distance between (0,0,0) and plane 2x+3y+6z=14 = (A) 1 (B) 2 (C) 14/7 (D) 14
31. If three vectors are coplanar, scalar triple is — (A) 1 (B) –1 (C) 0 (D) ±1
32. The line (x-1)/0 = (y-2)/1 = (z-3)/0 is parallel to — (A) x-axis (B) y-axis (C) z-axis (D) origin
33. Skew lines distance formula numerator — (A) [(a₂−a₁) b₁ b₂] (B) (a₂−a₁)·b₁ (C) b₁×b₂ (D) a₂−a₁
34. Equation of plane parallel to xy-plane at z=5 is — (A) x=5 (B) y=5 (C) z=5 (D) x+y=5
35. The angle between lines (1,1,1) and (-1,1,1) — (A) 0° (B) cos⁻¹(1/3) (C) 60° (D) 90°
36. If a·b = a·c and a≠0, then — (A) b=c (B) a⊥(b−c) (C) b∥c (D) b=–c
37. Equation of plane through (0,0,0) and (1,1,1), perpendicular to xy-plane — (A) x+y=0 (B) x-y=0 (C) z=0 (D) x+y+z=0
38. The plane 2x-3y+z=5 has direction cosines proportional to — (A) (2,-3,1) (B) (1,1,1) (C) (5,5,5) (D) (-2,3,-1)
39. Vector parallel to line of intersection of planes x+y=1 and y+z=1 — (A) (1,-1,1) (B) (1,1,1) (C) (-1,1,-1) (D) (0,0,0)
40. Cosine of angle between i+j+k and i (along x-axis) — (A) 0 (B) 1 (C) 1/√3 (D) 1/3

Answer Key

1-A, 2-B, 3-B, 4-B, 5-B, 6-A, 7-B, 8-B, 9-A, 10-B, 11-B, 12-B, 13-A, 14-A, 15-B, 16-B, 17-B, 18-D (for ⊥), 19-A, 20-A (because a×(a×a)=0), 21-A, 22-C, 23-A, 24-C, 25-B, 26-A, 27-B, 28-C, 29-A, 30-B, 31-C, 32-B, 33-A, 34-C, 35-B, 36-B, 37-B, 38-A, 39-A, 40-C

FAQ

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