Direction of Cosines and Direction Ratios
Theorem: If l, m, n are the direction cosines of a line L then l² + m² + n² = 1.
Proof: l = cosα = x/r
m = cosβ = y/r
n = cos γ = z/r
cos²α + cos²β + cos²γ = 1
(x/r) ² + (y/r) ² + (z/r) ² = 1
\(=\frac{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}{{{r}^{2}}}=\frac{{{r}^{2}}}{{{r}^{2}}}=1\).
l² + m² + n² = 1.
Theorem: The direction cosines of the line joining the point p(x₁, y₁, z₁), Q(x₂, y₂, z₂) are \(\left( \frac{{{x}_{2}}-{{x}_{1}}}{r},\frac{{{y}_{2}}-{{y}_{1}}}{r},\frac{{{z}_{2}}-{{z}_{1}}}{r} \right)\).
Where \(r=\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}+{{({{z}_{2}}-{{z}_{1}})}^{2}}}\).
Example: A line makes angle 90, 60 and 30 degrees with positive direction of x, y, z – axes respectively. Find the direction cosines.
Solution: Suppose l, m, n are the direction cosines of the line, then
l = cosα = cos 90 = 0
m = cosβ = cos60 = ½
and n = cos√ = cos30 = √3/2
Direction cosines of the line are (0, ½, √3/2).