Question

Find a vector of magnitude of 5 units parallel to the resultant of the vectors $\overrightarrow{a}=2\hat{i}+3\hat{j}-\hat{k}\mathrm{and}\overrightarrow{b}=\hat{i}-2\hat{j}+\widehat{k.}$

Answer 1

Given the position vectors $\overrightarrow{a}=2\hat{i}+3\hat{j}-\hat{k}$ and $\overrightarrow{b}=\hat{i}-2\hat{j}+\hat{k}$
∴ Resultant Vector = $\overrightarrow{a}+\overrightarrow{b}=2\hat{i}+3\hat{j}-\hat{k}+\hat{i}-2\hat{j}+\hat{k}=3\hat{i}+\hat{j}$
So, a unit vector parallel to the resultant vector is $\frac{3\hat{i}+\hat{j}}{\left|3\hat{i}+\hat{j}\right|}=\frac{3\hat{i}+\hat{j}}{\sqrt{3^2+1^2}}=\frac{3\hat{i}+\hat{j}}{\sqrt{10}}$
Hence, required vector = $5\times \frac{\left(3\hat{i}+\hat{j}\right)}{\sqrt{10}}=\sqrt{\frac{5}{2}}\left(3\hat{i}+\hat{j}\right)$