Question

If $\overline{a}=\overline{i}-3\overline{j}+2\overline{k},\overline{b}=2\overline{i}+\overline{j}-\overline{k}$ then the length of the component vector of $\overline{a}\times \overline{b}$ along $5\overline{i}-\overline{k}$ is

• A. $\sqrt{\frac{1}{13}}$
• B. $\sqrt{\frac{2}{13}}$
• C. $\sqrt{\frac{3}{13}}$
• D. $\sqrt{\frac{4}{13}}$

Answer 1

Answer: (B)

$\sqrt{\frac{2}{13}}$

$\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -3& 2\\ 2& 1& -1\end{array}\right|=\hat{i}+5\hat{j}+7\hat{k}$
Projection of $\overrightarrow{a}\times \overrightarrow{b}$ on $5\hat{i}-\hat{k}$
$=\frac{(\overrightarrow{a}\times \overrightarrow{b})(5\hat{i}-\hat{k})}{|5\hat{i}-\hat{k}|}$
$=\frac{2}{\sqrt{26}}$
$=\sqrt{\frac{2}{13}}$