# The direction cosines of the vector $(2\hat i+2\hat j-\hat k)$ are ___________.

$\begin{array}{1 1} (A)\;\bigg(\large\frac{2}{3},\frac{-2}{3},\frac{-1}{3}\bigg) \\ (B)\;\bigg(\large\frac{2}{3},\frac{2}{3},\frac{1}{3}\bigg) \\(C)\;\bigg(\large\frac{2}{3},\frac{2}{3},\frac{-1}{3}\bigg) \\(D)\;\bigg(\large\frac{2}{5},\frac{2}{5},\frac{-1}{5}\bigg) \end{array}$

Toolbox:
• If a directed line segement $OP$ makes angles $\alpha,\beta,\gamma$ with $OX,OY$ and $OZ$ respectively, then $\cos \alpha, \cos \beta, \cos \gamma$ are known as the direction cosines of $OP$
The direction cosines of the vector $(2 \hat i+2 \hat j-\hat k)$ can be determined as follows
Let $\overrightarrow a=2 \hat i+2 \hat j-\hat k$
The magnitude of $\overrightarrow a$ is $|\overrightarrow a|=\sqrt {2^2+2^2+(-1)^2}$
$|\overrightarrow a|=\sqrt 9=3$
Therefore direction cosines is $\bigg(\large\frac{2}{3},\frac{2}{3},\frac{-1}{3}\bigg)$