Direction Cosines from Direction Ratios

If (a, b, c) are direction ratios of a line, the direction cosines are obtained by dividing by √(a²+b²+c²). Direction ratios are proportional to direction cosines but not normalized.

Direction ratios $(a, b, c)$ of a line are any three numbers proportional to its direction cosines — i.e., $l:m:n = a:b:c$.

So $l = ka$, $m = kb$, $n = kc$ for some scalar $k$.

Using $l^2+m^2+n^2 = 1$:

$$k^2(a^2+b^2+c^2) = 1 \implies k = \pm\frac{1}{\sqrt{a^2+b^2+c^2}}$$

Therefore:

$$l = \frac{a}{\sqrt{a^2+b^2+c^2}}, \quad m = \frac{b}{\sqrt{a^2+b^2+c^2}}, \quad n = \frac{c}{\sqrt{a^2+b^2+c^2}}$$

(or the negatives of all three — for the opposite direction).

Key difference: Direction cosines are unique (up to an overall sign flip); direction ratios are not unique — any scalar multiple $(ka, kb, kc)$ represents the same line direction.

Example: Line joining $A(1,2,3)$ to $B(4,6,3)$: DRs are $(3, 4, 0)$. DCs: $l = 3/5$, $m = 4/5$, $n = 0$. ($\sqrt{9+16+0} = 5$.) The line is parallel to the xy-plane since $n = 0$.