3D Geometry: Distance & Angle Formulas JEE Main Guide

Three-Dimensional Geometry has been growing in JEE Main's Mathematics section, contributing 4–5 questions in recent sessions. The chapter rewards students with strong formula knowledge and three-dimensional visualisation — it is not conceptually deep but requires formula precision and the ability to set up vector or Cartesian equations correctly from a geometric description. This guide focuses on the distance and angle formulas that constitute the core of JEE's 3D testing.

Equations of Lines in 3D

A line through point (x₁,y₁,z₁) with direction ratios (a,b,c): Cartesian form: (x−x₁)/a = (y−y₁)/b = (z−z₁)/c = λ. Vector form: r = (x₁î+y₁ĵ+z₁k̂) + λ(aî+bĵ+ck̂) = a₀ + λd, where a₀ is the position vector of the point and d is the direction vector. Line through two points (x₁,y₁,z₁) and (x₂,y₂,z₂): direction ratios = (x₂−x₁, y₂−y₁, z₂−z₁). Direction cosines l, m, n: l = a/√(a²+b²+c²), m = b/√(a²+b²+c²), n = c/√(a²+b²+c²); and l²+m²+n² = 1 always. JEE tests: given the direction cosines, find the angle the line makes with coordinate axes (angle with x-axis = cos⁻¹(l), etc.)

Angle between two lines with direction ratios (a₁,b₁,c₁) and (a₂,b₂,c₂): cosθ = |a₁a₂+b₁b₂+c₁c₂|/[√(a₁²+b₁²+c₁²)·√(a₂²+b₂²+c₂²)]. Lines are perpendicular when a₁a₂+b₁b₂+c₁c₂ = 0. Lines are parallel when a₁/a₂ = b₁/b₂ = c₁/c₂. Take a free 3D Geometry mock to test your formula speed. For the full vectors and 3D framework, see our 3D geometry and vectors guide.

Equations of Planes and Key Distances

Plane equation: ax+by+cz+d=0. Normal vector to this plane: (a,b,c). Plane through point (x₁,y₁,z₁) with normal (a,b,c): a(x−x₁)+b(y−y₁)+c(z−z₁) = 0. Intercept form: x/p + y/q + z/r = 1 (cuts x-axis at p, y-axis at q, z-axis at r). Distance from point (x₀,y₀,z₀) to plane ax+by+cz+d=0: D = |ax₀+by₀+cz₀+d|/√(a²+b²+c²). Angle between two planes a₁x+b₁y+c₁z+d₁=0 and a₂x+b₂y+c₂z+d₂=0: cosθ = |a₁a₂+b₁b₂+c₁c₂|/[√(a₁²+b₁²+c₁²)·√(a₂²+b₂²+c₂²)]. Note: this is identical in form to the angle between two lines with those direction ratios — the normal vectors of the planes play the role of the direction vectors.

Distance between two parallel planes ax+by+cz+d₁=0 and ax+by+cz+d₂=0: D = |d₁−d₂|/√(a²+b²+c²). This result comes from taking any point on the first plane and computing its distance to the second. For line-plane intersection: set the parametric line equation into the plane equation to solve for λ. For the planes chapter, see our 3D planes and lines guide.

Skew Lines: Shortest Distance and Condition

Two lines in 3D are either: parallel (same direction), intersecting (meet at a point), or skew (neither parallel nor intersecting). For skew lines with vector equations r = a₁+λb₁ and r = a₂+μb₂: Shortest distance (SD) = |(a₂−a₁)·(b₁×b₂)| / |b₁×b₂|. Lines intersect if SD = 0, i.e., (a₂−a₁)·(b₁×b₂) = 0 (the scalar triple product is zero). Condition for coplanarity of two lines: same condition, i.e., (a₂−a₁)·(b₁×b₂) = 0. This is one of the most tested results in 3D JEE questions. For the detailed derivation and more examples, see our skew lines guide.

For Cartesian skew lines: r₁: (x−x₁)/a₁=(y−y₁)/b₁=(z−z₁)/c₁ and r₂: (x−x₂)/a₂=(y−y₂)/b₂=(z−z₂)/c₂. SD = |determinant of [(x₂−x₁, y₂−y₁, z₂−z₁), (a₁,b₁,c₁), (a₂,b₂,c₂)]| / √[(b₁c₂−b₂c₁)²+(c₁a₂−c₂a₁)²+(a₁b₂−a₂b₁)²]. The numerator is the scalar triple product, the denominator is |b₁×b₂|. JEE presents this as a 4-mark integer question: compute the exact shortest distance between two given skew lines. Practise computing 3×3 determinants rapidly to handle this in under 3 minutes.

Angle Between Line and Plane

Angle φ between a line with direction (a,b,c) and a plane with normal (p,q,r): sinφ = |ap+bq+cr|/[√(a²+b²+c²)·√(p²+q²+r²)]. Note: φ is the complement of the angle between the line and the plane's normal — hence sin instead of cos. A line is parallel to a plane when ap+bq+cr = 0 (direction perpendicular to normal, i.e., direction lies in the plane). A line is perpendicular to a plane when direction ∝ normal, i.e., a/p = b/q = c/r. Foot of perpendicular from a point to a plane: use the parametric line through the point in direction of normal, intersect with the plane. For the complete 3D Geometry chapter strategy, see our Math 2026 strategy guide.