Question

For given vectors, $a=2\hat{i}-\hat{j}+2\hat{k}andb=-\hat{i}+\hat{j}-\hat{k}$, find the unit vector in the direction of the vector $a+b$.

Answer 1

The given vectors are $a=2\hat{i}-j+2\hat{k}andb=-\hat{i}+\hat{j}-\hat{k}$
$\therefore a+b=(2\hat{i}-\hat{j}+2\hat{k})+(-\hat{i}+\hat{j}-\hat{k})$
Two vectors can be added by adding $\hat{i},\hat{j}and\hat{k}$ components.
$\therefore a+b=[2\hat{i}+(-\hat{i}\left)\right]+\left[\right(-\hat{j})+\hat{j}]+\left[\right(2\hat{k})+(-\hat{k}\left)\right]=(2\hat{i}-\hat{i})+(-\hat{j}+\hat{j})+(2\hat{k}-\hat{k})=\hat{i}+0\hat{j}+\hat{k}=\hat{i}+\hat{k}$
Comparing with $X=x\hat{i}+y\hat{j}+z\hat{k},wegetx=1,y=0,z=1$
$\therefore Magnitude|a+b|=\sqrt{x^2+y^2+z^2}=\sqrt{1^2+0^2+1^2}=\sqrt{2}$
Hence, the unit vector in the direction of $(a+b)$,
$\frac{(a+b)}{|a+b|}=\frac{(\hat{i}+\hat{k})}{\sqrt{2}}=\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{k}$