Three Dimensional Geometry | Class 12 Mathematics Notes

11.2 Direction Cosines and Direction Ratios of a Line

A directed line L passing through the origin in three-dimensional space makes angles α, β, and γ with the positive x, y, and z axes respectively. These angles are called direction angles. The cosines of these angles, namely cos α, cos β, and cos γ, are called the direction cosines of the line L. If the direction of the line is reversed, the direction angles become their supplements (π - α, π - β, π - γ), and the signs of the direction cosines change accordingly. To uniquely define direction cosines for a line (which can be extended in two opposite directions), the line must be considered as a directed line. The unique direction cosines are denoted by l, m, and n. If a line does not pass through the origin, a parallel line through the origin is considered to find its direction cosines, as parallel lines share the same direction cosines. Any three numbers proportional to the direction cosines are called direction ratios of the line. If l, m, n are direction cosines and a, b, c are direction ratios, then a = λl, b = λm, and c = λn for some nonzero scalar λ. The relation between direction cosines and direction ratios is given by l/a = m/b = n/c = k, where k is a constant. Using the property that l² + m² + n² = 1, k can be found as ±1/√(a² + b² + c²). Thus, direction cosines can be expressed as l = ±a/√(a² + b² + c²), m = ±b/√(a² + b² + c²), and n = ±c/√(a² + b² + c²).

📊 Diagram: Fig 11.1

🧪 Activity: Refer to the NCERT Mathematics Laboratory Handbook for activities involving direction cosines.

🔗 Connection: Prepares for calculating direction cosines of a line passing through two points.