# Direction of Cosines and Direction Ratios

## 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).