Three-Dimensional Geometry

Direction cosines (l, m, n): the cosines of the angles a line makes with positive x, y, z axes.

Direction ratios (a, b, c): any vector parallel to the line. (l, m, n) = (a, b, c) / √(a² + b² + c²).

Property: l² + m² + n² = 1.

EQUATIONS OF A LINE in 3D

Vector form: r⃗ = a⃗ + λ b⃗, where a⃗ is a point and b⃗ is direction.

Cartesian form:
(x − x₁) / a = (y − y₁) / b = (z − z₁) / c

This represents a line through (x₁, y₁, z₁) with direction ratios (a, b, c).

Through two points (x₁, y₁, z₁) and (x₂, y₂, z₂):
(x − x₁)/(x₂ − x₁) = (y − y₁)/(y₂ − y₁) = (z − z₁)/(z₂ − z₁)

Angle between two lines with direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂):

cos θ = (a₁a₂ + b₁b₂ + c₁c₂) / √[(a₁² + b₁² + c₁²)(a₂² + b₂² + c₂²)]

Lines are perpendicular iff a₁a₂ + b₁b₂ + c₁c₂ = 0.

Lines are parallel iff a₁/a₂ = b₁/b₂ = c₁/c₂.

EQUATIONS OF A PLANE

Vector form: r⃗ · n⃗ = d, where n⃗ is normal to the plane.

Cartesian form (general): ax + by + cz + d = 0.

• Normal vector: (a, b, c).
• Distance from origin: |d| / √(a² + b² + c²).

Plane through 3 points (P, Q, R): normal n⃗ = PQ⃗ × PR⃗.

Distance from a point (x₀, y₀, z₀) to plane ax + by + cz + d = 0:

Distance = |ax₀ + by₀ + cz₀ + d| / √(a² + b² + c²)

Angle between two planes with normals (a₁, b₁, c₁) and (a₂, b₂, c₂):

cos θ = (a₁a₂ + b₁b₂ + c₁c₂) / √[(a₁² + b₁² + c₁²)(a₂² + b₂² + c₂²)]

(Same as angle between two lines, applied to the normals.)

Angle between a line and a plane:

If line direction is (a, b, c) and plane normal is (l, m, n):

sin θ = (al + bm + cn) / √[(a² + b² + c²)(l² + m² + n²)]

Note the sin (because angle between line and plane = 90° − angle between line and normal).

Line is parallel to plane iff al + bm + cn = 0 (i.e., direction perpendicular to normal).

Worked example. Find the distance from the point (1, 2, 3) to the plane 2x − y + 2z + 5 = 0.

Distance = |2(1) − 1(2) + 2(3) + 5| / √(4 + 1 + 4) = |2 − 2 + 6 + 5| / 3 = 11/3 ≈ 3.67.

Worked example. Are the lines L₁: (x−1)/2 = (y+1)/3 = (z−2)/4 and L₂: (x+2)/3 = (y−1)/(−2) = z/1 perpendicular?

Direction ratios: (2,3,4) and (3,−2,1). Dot product = 6 − 6 + 4 = 4 ≠ 0. Not perpendicular.