# Find the unit vector in the direction of sum of vectors $\overrightarrow{a}=2\hat i+\hat j+\hat k\;and\;\overrightarrow{b}=2\hat j-\hat k$

$\begin{array}{1 1} (A)\;\Large\frac{2\hat i-\hat j-2\hat k }{\sqrt 3} \\ (B)\;\Large\frac{2\hat i-\hat j-2\hat k }{3}\\ (C)\;\Large\frac{2\hat i-\hat j-2\hat k }{9} \\ (D)\;\Large\frac{2\hat i-\hat j-2\hat k }{12}\end{array}$

Toolbox:
• The algebraic sum of the two vectors in the sum of the components of the two vectors.
• Unit vector in the direction of $\overrightarrow {a}$ is $\large\frac{ \overrightarrow {a}}{|\overrightarrow {a}|}$
$\overrightarrow{a}=2\hat i+\hat j+\hat k\;and\;\overrightarrow{b}=2\hat j-\hat k$
The sum of the vectors = $\overrightarrow {a}+ \overrightarrow {b}$
$=(2\hat i-\hat j-\hat k) +(2\hat j-\hat k)$
$\overrightarrow {a}+ \overrightarrow {b}=2\hat i-\hat j-2\hat k$
The magnitude of $|\overrightarrow {a}+ \overrightarrow {b}|=\sqrt {(2)^2+(-1)^2+(-2)^2}$
$\qquad \qquad \qquad \qquad \qquad =\sqrt {4+1+4}$
$\qquad \qquad \qquad \qquad \qquad =\sqrt {9}=3$
Hence the Unit vector in the direction of $\overrightarrow {a}+\overrightarrow {b}$ is $\large\frac{ \overrightarrow {a}+\overrightarrow {b}} {|\overrightarrow {a}+\overrightarrow {b}|}$
$=\Large\frac{2\hat i-\hat j-2\hat k }{3}$