Question

Direction Cosines and Direction Ratios of a Line of three dimensional geometry.

Answer 1

Direction Cosines and Direction Ratios of a Line:

Direction Cosines: If a directed line L makes angles α, β and γ with positive direction of X-axis, Y-axis and Z-axis respectively, then cos α, cos β and cos γ, are called direction cosines of a line. They are denoted by Z, m and n.

Therefore, l = cos a, m = cos β and n = cos γ Also, stun of squares of directed cosines of a line is always 1,

i.e., l² + m² + n² = 1

⇒ cos²α + cos²β + cos²γ = 1.

If we reverse the direction of L, then the direction angles are replaced by their supplements, i.e., n - α, n - β and it n - γ.

Thus, the signs of the direction cosines are reversed.

• Direction cosines of a directed line are unique.
• Direction cosines of X, Y and Z axes are (1, 0, 0), (0,1,0) and (0,0,1).
• If the given line in space does not pass through the origin, then, in order to find its direction cosines, we draw a line through the origin and parallel to the given line. Now take one of the directed lines from the origin and find its direction cosines as two parallel line have same set of direction cosines.