Question

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

• A. $\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}+\frac{4}{5\sqrt{2}}\hat{k}$
• B. $\frac{3}{5\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}+\frac{4}{5\sqrt{2}}\hat{k}$
• C. $\frac{3}{5\sqrt{10}}\hat{i}+\frac{1}{5\sqrt{10}}\hat{j}+\frac{4}{5\sqrt{10}}\hat{k}$
• D. $\frac{3}{5\sqrt{2}}\hat{i}-\frac{1}{\sqrt{2}}\hat{j}+\frac{4}{5\sqrt{2}}\hat{k}$

Answer 1

The correct option is B $\frac{3}{5\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}+\frac{4}{5\sqrt{2}}\hat{k}$

Let $\overrightarrow{a}$ be the resultant of given vectors.
Then $\overrightarrow{a}=(\hat{i}+2\hat{j}+3\hat{k})+(-\hat{i}+2\hat{j}+\hat{k})+(3\hat{i}+\hat{j})$
$=3\hat{i}+5\hat{j}+4\hat{k}$
$|\overrightarrow{a}|=\sqrt{(3{)}^2+(5{)}^2+(4{)}^2}$
$=\sqrt{9+25+16}=\sqrt{50}=5\sqrt{2}$
Now unit vector along $\overrightarrow{a}=\frac{\overrightarrow{a}}{|\overrightarrow{a}|}=\frac{3}{5\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}+\frac{4}{5\sqrt{2}}\hat{k}$