Introduction To Three- Dimensional Geometry

A 3D coordinate system consists

A 3D coordinate system consists of three mutually perpendicular axes (X, Y, Z) intersecting at origin O. Any point P in space is represented as P(x, y

Distance between points P(x₁

Distance between points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is |PQ| = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]. This extends the 2D Pythagorean theorem to three

For internal division

For internal division: Point dividing line segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) in ratio m:n internally is ((nx₂+mx₁)/(m+n), (ny₂+my₁)/(m+n

Direction cosines (l

Direction cosines (l, m, n) of a line are cosines of angles α, β, γ it makes with positive X, Y, Z axes respectively. Key property: l² + m² + n² = 1.

Projection of line segment PQ on

Projection of line segment PQ on a line with direction cosines (l, m, n) = (x₂-x₁)l + (y₂-y₁)m + (z₂-z₁)n, where P(x₁, y₁, z₁) and Q(x₂, y₂, z₂). Proj