CBSE Class 12 Maths: 3D Geometry — Complete Notes 2026

Direction Cosines and Direction Ratios

• Direction cosines (l, m, n): cosines of angles α, β, γ made with x, y, z axes; l² + m² + n² = 1
• Direction ratios (a, b, c): proportional to direction cosines; l = a/√(a²+b²+c²) etc.
• If direction ratios of PQ are (a, b, c), direction ratios of QP are (−a, −b, −c)

Equation of a Line in 3D

• Cartesian: (x−x₁)/a = (y−y₁)/b = (z−z₁)/c where (a,b,c) are direction ratios
• Vector form: r⃗ = a⃗ + λd⃗ where a⃗ is position vector of point, d⃗ is direction vector
• Find equation from two points: direction ratios = (x₂−x₁, y₂−y₁, z₂−z₁)

Angle Between Two Lines

• cos θ = |l₁l₂ + m₁m₂ + n₁n₂| (using direction cosines)
• Using direction ratios: cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / [√(a₁²+b₁²+c₁²) × √(a₂²+b₂²+c₂²)]
• Perpendicular lines: a₁a₂ + b₁b₂ + c₁c₂ = 0; Parallel lines: a₁/a₂ = b₁/b₂ = c₁/c₂

Equation of a Plane

• Normal form: ax + by + cz = d where (a,b,c) is normal to plane
• Plane through three points: set up 3×3 determinant = 0
• Intercept form: x/a + y/b + z/c = 1 (a, b, c are x, y, z intercepts)

Angle Between Line and Plane

• sin θ = |al + bm + cn| / [√(a²+b²+c²) × √(l²+m²+n²)] where (a,b,c) is normal, (l,m,n) is line direction
• Line parallel to plane: al + bm + cn = 0; Line perpendicular to plane: a/l = b/m = c/n
• Coplanarity of two lines: set up determinant with position difference and two direction vectors = 0

Distance Between Points, Lines, Planes

• Distance from point (x₀,y₀,z₀) to plane ax+by+cz+d=0: |ax₀+by₀+cz₀+d| / √(a²+b²+c²)
• Distance between parallel planes ax+by+cz+d₁=0 and ax+by+cz+d₂=0: |d₁−d₂|/√(a²+b²+c²)
• Foot of perpendicular from point to plane: parametric formula along normal direction

CBSE Board Focus

• 3D Geometry: 7–9 marks; line equations, angle between lines/planes, distance formula
• Plane through intersection of two planes: P₁ + λP₂ = 0; find λ using additional condition
• Equation of plane containing a line and perpendicular to another plane: frequently asked