Question

Find a vector of magnitude $6$ which is perpendicular to both the vectors $\overrightarrow{a}=4\hat{i}-\hat{j}+3\hat{k}$ and $\overrightarrow{b}=-2\hat{i}+\hat{j}-2\hat{k}$.

Answer 1

Given

$\overrightarrow{a}=4\hat{i}-\hat{j}+3\hat{k}$

$\overrightarrow{b}=-2\hat{i}+\hat{j}-2\hat{k}$

Calculate $\overrightarrow{a}\times \overrightarrow{b}$.

$\Rightarrow \left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 4& -1& 3\\ -2& 1& -2\end{array}\right|$

$\Rightarrow \hat{i}(2-3)-\hat{j}(-8+6)+\hat{k}(4-2)$

$\Rightarrow -\hat{i}+2\hat{j}+2\hat{k}$

Again,

$|\overrightarrow{a}\times \overrightarrow{b}|=\sqrt{1+4+4}=3$

Therefore, required vector is,

$=\frac{6}{3}(-\hat{i}+2\hat{j}+2\hat{k})$

$=-2\hat{i}+4\hat{j}+4\hat{k}$

Hence, this is the required result.