3D Geometry

Concept overview

A point in 3D is (x, y, z). A vector from origin to this point is xî + yĵ + zk̂. Direction ratios of a line are any triple proportional to its direction vector; direction cosines are the same triple normalised (l² + m² + n² = 1).

A line through (x₁, y₁, z₁) with direction ratios (a, b, c) has parametric form (x₁ + at, y₁ + bt, z₁ + ct) and symmetric form (x − x₁)/a = (y − y₁)/b = (z − z₁)/c.

A plane through (x₀, y₀, z₀) with normal (a, b, c) has equation a(x − x₀) + b(y − y₀) + c(z − z₀) = 0, or ax + by + cz + d = 0 in general form. Distance from point (x₁, y₁, z₁) to plane: |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²).

Angle between two lines with direction vectors u⃗ and v⃗ is cos θ = |u⃗ · v⃗|/(|u⃗| |v⃗|). Angle between two planes with normals n⃗₁, n⃗₂ has cos θ = |n⃗₁ · n⃗₂|/(|n⃗₁| |n⃗₂|). Distance between skew lines uses the shortest-distance formula involving cross-product of their direction vectors.

Key formulas & concepts

• Direction cosines: l² + m² + n² = 1
• Line symmetric form: (x − x₁)/a = (y − y₁)/b = (z − z₁)/c
• Plane general form: ax + by + cz + d = 0
• Distance point to plane: |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²)
• Angle between lines: cos θ = |u⃗ · v⃗|/(|u⃗| |v⃗|)
• Skew lines distance: |(r⃗₂ − r⃗₁) · (u⃗ × v⃗)|/|u⃗ × v⃗|

Common mistakes to avoid

• Confusing direction ratios (unnormalised) with direction cosines (normalised).
• Applying 2D angle formulas in 3D — need vector-based dot product.
• Forgetting the absolute value in cosine formulas for angle between lines.
• Using skew-lines distance when the lines are actually intersecting or parallel.

JEE Main relevance

JEE Main asks 1–2 3D problems every paper — usually one line-plane relation and one distance calculation.

JEE Advanced relevance

Advanced problems combine 3D geometry with vectors, asking about intersecting lines, image of a point in a plane, or foot of perpendicular from a point.

Previous-year questions (PYQ)

Distance from point to plane is the highest-frequency PYQ archetype.