Line and Plane in Space

Equations of a Plane

General equation: Ax + By + Cz + D = 0. The vector (A, B, C) is the normal vector n of the plane.

Normal equation: n·(r − r₀) = 0, where r₀ is a point of the plane, n is the normal.

Through three points: the 3×3 determinant composed of the differences with the first point equals zero.

Distance from point M(x₀, y₀, z₀) to the plane: d = |Ax₀ + By₀ + Cz₀ + D| / √(A²+B²+C²).

Angle between planes: cos φ = |n₁·n₂| / (|n₁||n₂|).

Equations of a Line

Parametric: r = r₀ + t·l, or x = x₀ + lt, y = y₀ + mt, z = z₀ + nt (l = (l, m, n) is the direction vector).

Symmetric: (x − x₀)/l = (y − y₀)/m = (z − z₀)/n.

As the intersection of two planes: A₁x + B₁y + C₁z + D₁ = 0, A₂x + B₂y + C₂z + D₂ = 0.

Relative Positions

Lines: intersect (coplanar, different directions), parallel (coplanar, same direction), skew (not coplanar).

Distance between skew lines: d = |(r₂ − r₁)·(l₁ × l₂)| / |l₁ × l₂|.

Line and plane: parallel (l ⟂ n and point does not lie), lies (l ⟂ n and point lies), intersects. Angle: sin φ = |l·n|/(|l||n|).