Coordinate Geometry in Three Dimensions
Three-dimensional coordinate geometry in the IPE papers covers direction cosines and direction ratios, the angle between lines, equations of planes in general, normal and intercept form, distance of a point from a plane, the angle between planes, and the symmetric form of a line together with line–plane intersection conditions.
Core rules and concepts
- Direction cosines l, m, n satisfy l² + m² + n² = 1; direction ratios a, b, c are proportional to them.
- Angle between lines with ratios (a₁, b₁, c₁) and (a₂, b₂, c₂): cos θ = (a₁a₂ + b₁b₂ + c₁c₂)/(√(a₁² + b₁² + c₁²)·√(a₂² + b₂² + c₂²)).
- Plane ax + by + cz + d = 0 has normal (a, b, c); intercept form x/α + y/β + z/γ = 1.
- Distance from (x₁, y₁, z₁) to ax + by + cz + d = 0 is |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²).
- Line (x − x₁)/a = (y − y₁)/b = (z − z₁)/c lies in the plane if ax₁ + by₁ + cz₁ + d = 0 and aa₁ + bb₁ + cc₁ = 0.
Worked example
Find the distance from (1, 2, 3) to the plane 2x − 3y + 6z = 21. Rewrite as 2x − 3y + 6z − 21 = 0, so a = 2, b = −3, c = 6, d = −21. Numerator: |2(1) − 3(2) + 6(3) − 21| = |2 − 6 + 18 − 21| = 7. Denominator: √(4 + 9 + 36) = 7. Distance = 7/7 = 1 unit.
Traps that cost marks
- Substituting into the distance formula without first moving the constant to the left side.
- Using direction ratios directly as direction cosines in a unit-vector question.
- Taking the angle between planes from their lines of intersection instead of their normals.
- Mishandling symmetric-form lines that contain a zero in the denominator.