Three Dimensional Geometry: One-Page Formula Sheet

Three Dimensional Geometry — formula sheet

Formulas

• Distance from point to plane: |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²)
• Shortest distance = |(b⃗ − a⃗)·(d₁ × d₂)| / |d₁ × d₂|
• Angle between planes: cosθ = |n₁·n₂|/(|n₁||n₂|)
• Foot of perpendicular via the parametric point

Key ideas

• l² + m² + n² = 1 for direction cosines
• Line: (x − x₁)/a = (y − y₁)/b = (z − z₁)/c
• Plane: ax + by + cz + d = 0 with normal (a, b, c)
• Skew lines are neither parallel nor intersecting

Traps to avoid

• Using direction ratios as cosines
• Sign of d in plane distance
• Forgetting absolute value in angles

jeevisionary.in/jee-topics/maths/three-d-geometry/