Question

What are the direction cosines of a line that passes through the points $P(6,-7,-1)$ and $Q(2,-3,1)$

Answer 1

Direction ratio of $\overrightarrow{PQ}=-4\hat{i}+4\hat{j}+2\hat{k}$

Now,

Direction cosine $=\frac{-4\hat{i}+4\hat{j}+2\hat{k}}{\sqrt{{\left(-4\right)}^2+{\left(4\right)}^2+{\left(2\right)}^2}}$

$=\frac{-4\hat{i}+4\hat{j}+2\hat{k}}{6}$

$=\frac{-2}{3}\hat{i}+\frac{2}{3}\hat{j}+\frac{1}{3}\hat{k}$

Hence, the direction cosine is $\left(\frac{-2}{3},\frac{2}{3},\frac{1}{3}\right)$