Three-Dimensional Geometry
Questions cover direction cosines and ratios, equations of lines and planes in vector and Cartesian form, angles, and shortest distances. Expect a 1-mark direction-ratio item and a 4–6 mark problem on planes or skew lines.
Standard forms
- Line through a with direction b: r = a + λb; Cartesian (x − x₁)/a = (y − y₁)/b = (z − z₁)/c.
- Plane with normal n through a: (r − a)·n = 0; Cartesian ax + by + cz = d has normal (a, b, c).
- Angle between two lines: cos θ = |b₁·b₂|/(|b₁||b₂|).
- Angle between a line and a plane: sin θ = |b·n|/(|b||n|), since θ is the complement of the angle with the normal.
- Distance of a point (x₁, y₁, z₁) from ax + by + cz + d = 0 is |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²).
- Shortest distance between skew lines with directions b₁, b₂: |(a₂ − a₁)·(b₁ × b₂)|/|b₁ × b₂|.
Worked example
Distance of (1, 2, 3) from 2x − 3y + 6z + 1 = 0 is |2(1) − 3(2) + 6(3) + 1|/√(4 + 9 + 36) = |2 − 6 + 18 + 1|/7 = 15/7 units.
Traps
- Using the angle with the normal as the angle between line and plane.
- Forgetting the modulus in the distance formula.
- Confusing direction ratios with direction cosines (cosines must satisfy l² + m² + n² = 1).
- Writing the normal vector wrongly from the Cartesian plane equation.