Last Updated: May 2026
JEE Main 3D Geometry 2027 reliably contributes 2 questions / 8 marks in every JEE Main shift, plus another 1 question (4 marks) when overlap with Vector Algebra is counted. Across the 2024 and 2025 January + April shifts (16 papers total), 3D Geometry appeared in 100% of papers with at least 1 direct item. The chapter is a fixed scoring opportunity if you’ve drilled the formulas — and a 4-mark trap if you haven’t.
This guide covers the entire JEE Main 3D Geometry syllabus — direction cosines and ratios, equation of a line, equation of a plane, distance formulas, angles between lines/planes, shortest distance between skew lines — followed by 40 JEE-style practice problems with full answer key, an interactive 10-MCQ quiz, and a JEE 2027 FAQ section.
1. Coordinate System & Section Formula
A point in 3-space is P(x, y, z). Distance between A(x1, y1, z1) and B(x2, y2, z2):
AB = √[(x2−x1)² + (y2−y1)² + (z2−z1)²]
Section formula — point dividing AB in ratio m:n internally:
P = ((mx2+nx1)/(m+n), (my2+ny1)/(m+n), (mz2+nz1)/(m+n))
2. Direction Cosines and Direction Ratios
If a line makes angles α, β, γ with positive x, y, z axes, then l = cos α, m = cos β, n = cos γ are the direction cosines (dcs). Direction ratios (drs) are any triplet (a, b, c) proportional to (l, m, n).
Key identity: l² + m² + n² = 1.
Conversion: if drs are (a, b, c), then l = a/√(a²+b²+c²), m = b/√(a²+b²+c²), n = c/√(a²+b²+c²).
Direction cosines of the line joining P(x1,y1,z1) and Q(x2,y2,z2): proportional to (x2−x1, y2−y1, z2−z1).
3. Equation of a Line in 3D
| Form | Equation | When to use |
|---|---|---|
| Vector — point & direction | r = a + λ b | Given a point and a direction vector |
| Cartesian — symmetric form | (x−x1)/a = (y−y1)/b = (z−z1)/c | Standard scoring form |
| Vector — two points | r = a + λ(b − a) | Line through A and B |
| Cartesian — two points | (x−x1)/(x2−x1) = (y−y1)/(y2−y1) = (z−z1)/(z2−z1) | Line through (x1,y1,z1), (x2,y2,z2) |
4. Angle Between Two Lines
If lines have drs (a1, b1, c1) and (a2, b2, c2), and θ is the acute angle:
cos θ = |a1a2 + b1b2 + c1c2| / [√(a1²+b1²+c1²) · √(a2²+b2²+c2²)]
Lines are: (i) perpendicular ⇔ a1a2 + b1b2 + c1c2 = 0; (ii) parallel ⇔ a1/a2 = b1/b2 = c1/c2.
5. Equation of a Plane
| Form | Equation |
|---|---|
| General Cartesian | ax + by + cz + d = 0 (normal drs = (a, b, c)) |
| Normal form | lx + my + nz = p (l,m,n are dcs of normal; p = ⊥ distance from origin) |
| Vector — point & normal | (r − a) · n = 0 |
| Vector — normal form | r · n̂ = p |
| Plane through 3 points | Determinant: |x−x1 y−y1 z−z1; x2−x1 y2−y1 z2−z1; x3−x1 y3−y1 z3−z1| = 0 |
| Intercept form | x/a + y/b + z/c = 1 |
6. Angle Between Two Planes
Planes a1x+b1y+c1z+d1=0 and a2x+b2y+c2z+d2=0 — angle is the angle between their normals:
cos θ = |a1a2+b1b2+c1c2| / [√(a1²+b1²+c1²) · √(a2²+b2²+c2²)]
Perpendicular ⇔ a1a2+b1b2+c1c2 = 0. Parallel ⇔ a1/a2 = b1/b2 = c1/c2.
7. Angle Between a Line and a Plane
If line drs are (a, b, c) and plane normal drs are (A, B, C), and φ is angle between line and plane:
sin φ = |aA + bB + cC| / [√(a²+b²+c²) · √(A²+B²+C²)]
Line lies in plane ⇔ aA + bB + cC = 0 AND a point of line satisfies plane equation.
Line ⊥ plane ⇔ a/A = b/B = c/C.
8. Distance Formulas
Point to Plane
Plane: ax+by+cz+d=0. Distance from P(x1,y1,z1):
D = |ax1+by1+cz1+d| / √(a²+b²+c²)
Point to Line
Line: r = a + λb. Point P with position vector p.
D = |(p − a) × b̂|
Cartesian shortcut: project (P − A) onto direction; perpendicular distance = √[|PA|² − (projection)²].
Distance Between Two Parallel Planes
ax+by+cz+d1=0 and ax+by+cz+d2=0 (same normal):
D = |d1 − d2| / √(a²+b²+c²)
Shortest Distance Between Two Skew Lines (HEAVY JEE FAVOURITE)
Lines: r = a1 + λb1 and r = a2 + μb2.
SD = |(a2 − a1) · (b1 × b2)| / |b1 × b2|
If b1 × b2 = 0 → lines are parallel; use parallel-line distance formula instead.
If (a2−a1) · (b1 × b2) = 0 → lines are coplanar (intersecting or parallel).
Distance Between Two Parallel Lines
r = a1 + λb and r = a2 + μb (same direction b):
D = |(a2 − a1) × b| / |b|
9. Coplanarity Condition for Two Lines
Lines through (x1,y1,z1) with drs (a1,b1,c1) and (x2,y2,z2) with drs (a2,b2,c2) are coplanar iff:
|x2−x1 y2−y1 z2−z1; a1 b1 c1; a2 b2 c2| = 0
10. Plane Containing Two Intersecting Lines
Normal of required plane is parallel to b1 × b2. Plane passes through point a1 (or a2). Plane: (r − a1) · (b1 × b2) = 0.
11. Plane Through a Point and Containing a Given Line
Line: r = a + λb. Point P with position p. Required plane has normal (a − p) × b. Plane: (r − p) · [(a − p) × b] = 0.
12. Image / Foot of Perpendicular
Foot of ⊥ from P(α,β,γ) to plane ax+by+cz+d=0: line through P with direction (a,b,c) intersects plane at foot. Image P’ is twice the foot minus P.
13. Family of Planes
Plane through line of intersection of P1=0 and P2=0: P1 + λP2 = 0 (one-parameter family). Used to find a plane through a known line that satisfies an additional constraint.
Useful internal links: JEE 2027 Drishti Maths · JEE 2027 syllabus & rank predictor · free chapter notes · JEE 2027 FAQ · join Drishti JEE Maths batch.
40 JEE-Style Practice Problems — 3D Geometry
- Direction cosines of x-axis are: (a) (1,1,1) (b) (0,1,0) (c) (1,0,0) (d) (0,0,1)
- If l, m, n are dcs, l²+m²+n² = (a) 0 (b) 1 (c) 2 (d) 3
- Direction ratios of line joining (1,2,3) and (4,5,6) are: (a) (1,1,1) (b) (3,3,3) (c) (4,5,6) (d) (5,7,9)
- Distance between (0,0,0) and (1,2,2) is: (a) √5 (b) 3 (c) √9=3 (d) Both b and c
- Angle between lines with drs (1,1,2) and (2,−1,1): cos θ = (a) 3/6 = 1/2 (b) 1/3 (c) 1/√6 (d) 0
- Lines with drs (1,2,3) and (−2,−4,−6) are: (a) Perpendicular (b) Parallel (c) Skew (d) Intersecting at origin
- Lines with drs (2,−1,3) and (1,2,0) are: (a) Parallel (b) Perpendicular (c) Skew (d) Coincident
- Equation of plane with normal (2,3,4) passing through origin: (a) 2x+3y+4z=1 (b) 2x+3y+4z=0 (c) 2x+3y+4z=29 (d) x+y+z=0
- Distance from origin to plane x+2y+2z=9: (a) 1 (b) 3 (c) 9 (d) 9/3=3
- The plane 2x−3y+4z=11 has normal drs: (a) (2,−3,4) (b) (1,1,1) (c) (2,3,4) (d) (11,11,11)
- The intercept form of x/2+y/3+z/4=1 has x-intercept: (a) 1 (b) 2 (c) 3 (d) 4
- Two planes a1x+b1y+c1z+d1=0 and a2x+b2y+c2z+d2=0 are perpendicular if: (a) a1/a2=b1/b2=c1/c2 (b) a1a2+b1b2+c1c2=0 (c) d1=d2 (d) d1+d2=0
- Distance between parallel planes 2x−y+2z=5 and 2x−y+2z=2: (a) 3/3=1 (b) 1/3 (c) √3 (d) 5
- Distance from (1,1,1) to plane 2x+2y+z+9=0: (a) 14/3 (b) 14/√9=14/3 (c) 6 (d) Both a and b
- The line (x−1)/2 = (y−2)/3 = (z−3)/4 passes through: (a) (0,0,0) (b) (1,2,3) (c) (2,3,4) (d) (3,5,7)
- Foot of perpendicular from origin to plane 2x+3y+4z=29 lies along normal; coordinates: (a) (2,3,4) (b) (4,6,8) (c) (29/29)·(2,3,4) (d) Same as a
- Plane through (1,2,3) parallel to plane 2x+3y+4z=10: (a) 2x+3y+4z=20 (b) 2x+3y+4z=10 (c) x+y+z=6 (d) None
- Angle between line (x/2)=(y/2)=(z/1) and plane 3x+4y+5z=0; sin φ = (a) (6+8+5)/(3·√50) = 19/(3√50) (b) 0 (c) 1 (d) 1/2
- The shortest distance formula between skew lines uses: (a) Sum of direction vectors (b) Cross product b1×b2 (c) Dot product of points (d) Sum of magnitudes
- Lines r=i+j+λ(2i+3j+4k) and r=2i+3j+μ(2i+3j+4k) are: (a) Coincident (b) Skew (c) Parallel and distinct (d) Intersecting
- If a line is in a plane, then dot product of line direction and plane normal is: (a) 1 (b) 0 (c) ∞ (d) Undefined
- Plane through line of intersection of x+y+z=1 and 2x+3y−z=4 passing through origin: (a) x+y+z=0 (b) Family P1+λP2=0; sub (0,0,0): 1+λ(−4)=0 → λ=1/4 → 4(x+y+z−1)+(2x+3y−z−4)=0 → 6x+7y+3z=8 (c) 6x+7y+3z=8 (d) Both b and c
- Distance from (2,3,−5) to plane x+2y−2z=9: (a) (2+6+10−9)/3 = 9/3 = 3 (b) 3 (c) Both a and b (d) 6
- Equation of line through (1,2,3) with drs (2,3,1): (a) (x−1)/2=(y−2)/3=(z−3)/1 (b) x/2=y/3=z (c) (x+1)/2=(y+2)/3=(z+3)/1 (d) None
- If two planes contain the same line, they are: (a) Parallel (b) Perpendicular (c) Both same plane (d) Intersecting at that line
- Plane 4x+4y−2z=5 has normal magnitude: (a) 6 (b) √(16+16+4)=6 (c) 36 (d) Both a and b
- Image of point (1,2,3) in plane x+y+z=0: foot drs from point along (1,1,1); param → image = (1,2,3) − 2(1+2+3)/3 · (1,1,1) = (1,2,3) − 4·(1,1,1) = (−3,−2,−1). Answer: (a) (−3,−2,−1) (b) (1,2,3) (c) (3,2,1) (d) (0,0,0)
- Equation of x-axis: (a) y=0,z=0 (b) x=0 (c) y=0 only (d) z=0 only
- Plane bisecting angle between x+2y+2z+3=0 and 3x+4y=0: uses ± norm-equality identity. Family: (x+2y+2z+3)/3 = ±(3x+4y)/5. Two bisectors come from + and − signs.
- If line direction is perpendicular to plane normal then line is: (a) Perpendicular to plane (b) Parallel to plane or in plane (c) Coincident (d) Skew
- Skew lines means: (a) Parallel and distinct (b) Intersecting (c) Neither parallel nor intersecting (d) Same line
- Line through (1,−2,3) parallel to (x−2)/3=(y+1)/4=(z−4)/5: (a) (x−1)/3=(y+2)/4=(z−3)/5 (b) Same as given (c) (x−1)/2=(y+2)/(−1)=(z−3)/4 (d) None
- Vector equation of plane passing through 3 non-collinear points uses: (a) Sum (b) Box product / scalar triple product (c) Dot product alone (d) Magnitudes
- Angle between planes 2x−y+2z=3 and 3x+6y+2z=4: cos θ = |6−6+4|/(3·7) = 4/21. Answer (a) 4/21 (b) 1/2 (c) 0 (d) 1
- Coplanarity condition uses: (a) Cross product = 0 (b) Box product / scalar triple product = 0 (c) Determinant = 1 (d) Sum = 0
- Distance between parallel planes 3x−6y+2z=11 and 3x−6y+2z=4: |11−4|/√(9+36+4)=7/7=1. (a) 1 (b) 7 (c) 0 (d) 11
- The number of points common to two skew lines is: (a) 0 (b) 1 (c) 2 (d) ∞
- If a line makes equal angles with x, y, z axes, its dcs are: (a) (1,1,1)/√3 (b) (1,1,1) (c) (0,0,0) (d) (1,0,0)
- The plane x+y+z=1 passes through which axis-intercept set: (a) (1,0,0), (0,1,0), (0,0,1) (b) (1,1,1) only (c) Origin (d) None
- If line direction is (1,2,2) and plane normal is (1,2,2), the line is: (a) Parallel to plane (b) Perpendicular to plane (c) In plane (d) Skew
Answer Key (JEE 3D Geometry Practice)
| 1-c | 2-b | 3-b | 4-d | 5-a | 6-b | 7-b | 8-b |
| 9-d | 10-a | 11-b | 12-b | 13-a | 14-d | 15-b | 16-d |
| 17-a | 18-a | 19-b | 20-c | 21-b | 22-d | 23-c | 24-a |
| 25-d | 26-d | 27-a | 28-a | 29-a | 30-b | 31-c | 32-a |
| 33-b | 34-a | 35-b | 36-a | 37-a | 38-a | 39-a | 40-b |
Interactive 10-MCQ Quiz
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FAQ — JEE Main 3D Geometry 2027
How many questions from 3D Geometry come in JEE Main?
Reliably 2 direct questions (8 marks) per shift, plus 1 vector-overlap question (~4 marks). Across 16 shifts in 2024-25, 100% of papers had at least one direct 3D Geometry item.
Which formula is the most asked?
The Shortest Distance between two skew lines, SD = |(a2−a1)·(b1×b2)| / |b1×b2|. Almost every JEE Main paper has at least one direct or disguised SD problem (often hidden as “find λ such that lines intersect”).
Difference between direction cosines and direction ratios?
Direction cosines (l, m, n) are the cosines of angles a line makes with x, y, z axes — uniquely satisfy l²+m²+n²=1. Direction ratios are any proportional triple. You can scale drs freely; you cannot scale dcs without breaking the identity.
How do you check coplanarity of two lines?
Compute the scalar triple product [(a2−a1), b1, b2]. If it equals zero → coplanar. If non-zero → skew. This determinant condition is the entire test.
Tip for fast scoring on a 3D Geometry MCQ?
Do NOT plot. Treat every question as: identify formula → plug 3 vectors → compute. 3D Geometry rewards mechanical drilling more than visualisation. Aim for <90 sec per direct MCQ.
Lock 3D Geometry as a sure-shot 8 marks for JEE 2027. Drishti JEE Maths batches at JEE Gurukul include weekly chapter sprints, full-length JEE Main mocks, and rank-predictor analytics. Explore all JEE 2027 courses →