Question

If $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}-\hat{j}+3\hat{k}\mathrm{and}\overrightarrow{c}=\hat{i}-2\hat{j}+\hat{k},$find a unit vector parallel to $2\overrightarrow{a}-\overrightarrow{b}+3\overrightarrow{c.}$

Answer 1

We have, $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$, $\overrightarrow{b}=2\hat{i}-\hat{j}+3\hat{k}$ and $\overrightarrow{c}=\hat{i}-2\hat{j}+\hat{k}$
∴ $2\overrightarrow{a}-\overrightarrow{b}+3\overrightarrow{c}=2\left(\hat{i}+\hat{j}+\hat{k}\right)-\left(2\hat{i}-\hat{j}+3\hat{k}\right)+3\left(\hat{i}-2\hat{j}+\hat{k}\right)=3\hat{i}-3\hat{j}+2\hat{k}.$

A unit vector parallel to $2\overrightarrow{a}-\overrightarrow{b}+3\overrightarrow{c}$ is given by $\frac{2\overrightarrow{a}-\overrightarrow{b}+3\overrightarrow{c}}{\left|2\overrightarrow{a}-\overrightarrow{b}+3\overrightarrow{c}\right|}$$=\frac{\left(3\hat{i}-3\hat{j}+2\hat{k}\right)}{\sqrt{3^2+{\left(-3\right)}^2+2^2}}$
$$\begin{gathered} =\frac{\left(3\hat{i}-3\hat{j}+2\hat{k}\right)}{\sqrt{22}} \\ =\frac{3}{\sqrt{22}}\hat{i}-\frac{3}{\sqrt{22}}\hat{j}+\frac{2}{\sqrt{22}}\hat{k} \end{gathered}$$