Question

Find a unit vector in the direction of the resultant of the vectors $\hat{i}-\hat{j}+3\hat{k},2\hat{i}+\hat{j}-2\hat{k}\mathrm{and}\hat{i}+2\hat{j}-2\hat{k}.$

Answer 1

Given: $\overrightarrow{a}=\hat{i}-\hat{j}+3\hat{k},\overrightarrow{b}=2\hat{i}+\hat{j}-2\hat{k}$ and $\overrightarrow{c}=\hat{i}+2\hat{j}-2\hat{k}$ are the position vectors.
Then, Resultant of the vectors = $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}$
$$\begin{gathered} =\hat{i}-\hat{j}+3\hat{k}+2\hat{i}+\hat{j}-2\hat{k}+\hat{i}+2\hat{j}-2\hat{k} \\ =4\hat{i}+2\hat{j}-\hat{k} \end{gathered}$$

So, $\left|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right|=\sqrt{4^2+2^2+1^2}=\sqrt{16+4+1}=\sqrt{21}$
∴ Unit vector in the direction of the resultant vector = $\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{\left|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right|}=\frac{1}{\sqrt{21}}\left(4\hat{i}+2\hat{j}-\hat{k}\right)$