Question

If a line has the direction ratios $-18,12,-4$ then what are its direction cosines?

Answer 1

If the direction ratios of a line are $a,b,c$, then the direction cosines are:
$[\frac{a}{\sqrt{a^2+b^2+c^2}},\frac{b}{a^2+b^2+c^2},\frac{c}{\sqrt{a^2+b^2+c^2}}]$

Given direction ratios are:$-18,12,-4$

i.e. $a=-18,b=12,c=-4$

$Now,\sqrt{a^2+b^2+c^2}=\sqrt{(-18{)}^2+{12}^2+(-4{)}^2}$
$\Rightarrow \sqrt{a^2+b^2+c^2}=\sqrt{324+144+16}=\sqrt{484}$
$\Rightarrow \sqrt{a^2+b^2+c^2}=22$
$\therefore$ Direction cosines are:$\frac{-18}{22},\frac{12}{22},\frac{-4}{22}$
or$\frac{-9}{11},\frac{6}{11},\frac{-2}{11}$