Three Dimensional Geometry

Direction Cosines and Direction Ratios

Direction cosines are the cosines of the angles made by a line with the positive x, y, and z axes, denoted by (l, m, n) where l^2 + m^2 + n^2 = 1. Direction ratios are numbers proportional to the direction cosines, represented as (a, b, c).

Equation of a Line

A line in 3D space can be defined uniquely by passing through a fixed point with position vector a and parallel to a vector b, given in vector form as r = a + lambda*b.

Equation of a Plane

A plane can be determined by a normal vector perpendicular to the plane and a point on it, expressed in vector form as r dot n = p, or in Cartesian form as ax + by + cz = d.

Angle between Lines and Planes

The angle between two lines is determined by the dot product of their direction vectors, while the angle between a line and a plane uses the sine of the angle derived from the line's direction vector and the plane's normal.

Shortest Distance between Skew Lines

Skew lines are non-coplanar lines that neither intersect nor are parallel. The shortest distance formula between them utilizes vector cross products and scalar triple products.