3D Geometry: Lines & Planes in JEE Main 2026

Three-dimensional geometry is one of the highest-weightage chapters in JEE Main Mathematics, consistently contributing two to three questions per session. The chapter tests direction cosines and ratios, equations of lines and planes, angles between them, and shortest distance between skew lines. Mastering the vector and Cartesian forms of lines and planes, and understanding when to use each, is the core skill this chapter develops.

Direction Cosines and Direction Ratios

Direction cosines (l, m, n) of a line are the cosines of the angles the line makes with the positive x, y, z axes respectively. Key property: l² + m² + n² = 1. Direction ratios (a, b, c) are any multiples of direction cosines — they point in the same direction but need not satisfy a² + b² + c² = 1. Converting DRs to DCs: l = a/√(a²+b²+c²), m = b/√(a²+b²+c²), n = c/√(a²+b²+c²).

Angle between two lines: cosθ = |l₁l₂ + m₁m₂ + n₁n₂| (using DCs) = |a₁a₂ + b₁b₂ + c₁c₂|/√(a₁²+b₁²+c₁²) × √(a₂²+b₂²+c₂²) (using DRs). Lines are perpendicular: a₁a₂ + b₁b₂ + c₁c₂ = 0. Lines are parallel: a₁/a₂ = b₁/b₂ = c₁/c₂. JEE Main tests these conditions in the context of determining relationships between two given lines. For the vector algebra that underpins 3D geometry, see our Vector Algebra Guide.

Equations of Lines in 3D

Cartesian form: (x−x₁)/a = (y−y₁)/b = (z−z₁)/c = λ, where (x₁,y₁,z₁) is a point on the line and (a,b,c) are DRs of the direction. Vector form: r⃗ = a⃗ + λb⃗, where a⃗ is the position vector of a point and b⃗ is the direction vector. JEE Main gives a line in one form and asks questions requiring the other form — be comfortable converting between them. Take a free mock test on 3D geometry to practise converting between line forms and computing distances.

Equations of Planes

General form: ax + by + cz + d = 0, where (a,b,c) is the normal to the plane. Normal form: lx + my + nz = p (DCs of normal, p = perpendicular distance from origin). Point-normal form: a(x−x₁) + b(y−y₁) + c(z−z₁) = 0. Three-point form: use determinant. Intercept form: x/α + y/β + z/γ = 1. Vector form: (r⃗ − a⃗)·n⃗ = 0.

Angle between two planes: cosθ = |a₁a₂ + b₁b₂ + c₁c₂|/√(a₁²+b₁²+c₁²) × √(a₂²+b₂²+c₂²) (same formula as angle between lines using normal vectors). Planes are perpendicular when dot product of normals = 0. Planes are parallel when normals are proportional. Distance from point (x₁,y₁,z₁) to plane ax+by+cz+d=0: D = |ax₁+by₁+cz₁+d|/√(a²+b²+c²).

Angle Between Line and Plane

sinθ = |al + bm + cn|/√(a²+b²+c²) × √(l²+m²+n²), where (a,b,c) is the normal to the plane and (l,m,n) are DRs of the line. Note: the angle between a line and plane is the complement of the angle between the line and the plane's normal. Line is perpendicular to plane when (a,b,c) ∥ (l,m,n): l/a = m/b = n/c. Line is parallel to plane when (a,b,c) ⊥ (l,m,n): al + bm + cn = 0.

Shortest Distance Between Skew Lines

Two lines in 3D that are not parallel and do not intersect are called skew lines. Shortest distance between skew lines r⃗ = a₁⃗ + λb₁⃗ and r⃗ = a₂⃗ + μb₂⃗ is: SD = |(a₂⃗ − a₁⃗)·(b₁⃗ × b₂⃗)|/|b₁⃗ × b₂⃗|. The cross product b₁⃗ × b₂⃗ gives the common perpendicular direction. If this expression equals zero, the lines are coplanar (either intersecting or parallel). JEE Main tests shortest distance in both vector and Cartesian forms — the Cartesian form uses the determinant formula.

Exam Strategy

3D geometry questions are systematic — the formula set is well-defined and the execution is mechanical. The most common errors are: (1) using DRs instead of DCs in formulas that require DCs; (2) computing cross product incorrectly; (3) sign errors in plane equations. Practise the shortest-distance formula ten times until computing b₁⃗ × b₂⃗ and the dot product are automatic. For the full coordinate geometry treatment, see our Coordinate Geometry Guide. Upgrade for ₹149/month for 150+ 3D geometry problems with step-by-step vector solutions.