# Find the projection of $$\overrightarrow a = 2\hat i +3\hat j + 2\hat k\: on \: \overrightarrow b = \hat i + 2\hat i - \hat j + \hat k$$

Toolbox:
• The projection of vector $\overrightarrow a$ on vector $\overrightarrow b$ is given by $\large\frac{\overrightarrow a.\overrightarrow b}{\mid \overrightarrow b\mid}$
• $\hat i.\hat i=1,\hat j.\hat j=1,\hat k.\hat k=1$
Step 1:
Let $\overrightarrow a=2\hat i+3\hat j+2\hat k$
$\quad\;\overrightarrow b=\hat i+2\hat j+\hat k$
The projection of vector $\overrightarrow a$ on vector $\overrightarrow b$ is given by $\large\frac{\overrightarrow a.\overrightarrow b}{\mid \overrightarrow b\mid}$-----(1)
$\overrightarrow a.\overrightarrow b=(2\hat i+3\hat j+2\hat k).(\hat i+2\hat j+\hat k)$
$\hat i.\hat i=1,\hat j.\hat j=1,\hat k.\hat k=1$
$\qquad\;=2+6+2=10$
Step 2:
$\mid b\mid=\sqrt{1^2+2^2+1^2}$
$\qquad=\sqrt{1+4+1}$
$\qquad=\sqrt{6}$
Substituting these in equ(1) we get,
$\large\frac{\overrightarrow a.\overrightarrow b}{\mid \overrightarrow b\mid}=\large\frac{10}{\sqrt 6}$
$\qquad\quad=\large\frac{10\times \sqrt 6}{\sqrt 6 \times \sqrt 6}$
$\qquad\quad=\large\frac{10\times \sqrt 6}{6}$
$\qquad\quad=\large\frac{5\sqrt 6}{3}$