Question:
For given vectors, $\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}=-\hat{i}+\hat{j}-\hat{k}$, find the unit vector in the direction of the vector $\vec{a}+\vec{b}$
Solution:
The given vectors are $\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}=-\hat{i}+\hat{j}-\hat{k}$.
$\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}$
$\vec{b}=-\hat{i}+\hat{j}-\hat{k}$
$\therefore \vec{a}+\vec{b}=(2-1) \hat{i}+(-1+1) \hat{j}+(2-1) \hat{k}=1 \hat{i}+0 \hat{j}+1 \hat{k}=\hat{i}+\hat{k}$
$|\vec{a}+\vec{b}|=\sqrt{1^{2}+1^{2}}=\sqrt{2}$
Hence, the unit vector in the direction of $(\vec{a}+\vec{b})$ is
$\frac{(\vec{a}+\vec{b})}{|\vec{a}+\vec{b}|}=\frac{\hat{i}+\hat{k}}{\sqrt{2}}=\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{k}$