Question

A unit vector perpendicular to each of the vectors $2\hat{i}+4\hat{j}-\hat{k}$ and $\hat{i}-2\hat{j}+3\hat{k}$ forming a right handed system is

• A. $7\hat{i}-10\hat{j}+8\hat{k}$
• B. $\frac{10\hat{i}-7\hat{j}-8\hat{k}}{\sqrt{213}}$
• C. $-7\hat{i}+10\hat{j}+8\hat{k}$
• D. $\frac{-10\hat{i}-7\hat{j}-8\hat{k}}{\sqrt{213}}$

Answer 1

The correct option is B $\frac{10\hat{i}-7\hat{j}-8\hat{k}}{\sqrt{213}}$

Unit vector perpendicular to the given 2 vectors is $(2\hat{i}+4\hat{j}-\hat{k})\times (\hat{i}-2\hat{j}+3\hat{k})$
$=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 4& -1\\ 1& -2& 3\end{array}\right|$
$=\hat{i}(12-2)-\hat{j}(6+1)+\hat{k}(-4-4)$
$=10\hat{i}-7\hat{j}-8\hat{k}$
The unit vector will be obtained by dividing the obtained vector with its modulus.
So, we get the answer as $\frac{10\hat{i}-7\hat{j}-8\hat{k}}{\sqrt{100+49+64}}=\frac{10\hat{i}-7\hat{j}-8\hat{k}}{\sqrt{213}}$