Question

Find a vector of magnitude 6, perpendicular to each of the vectors $\underset{a}{\to }+\underset{b}{\to }$ and $\underset{a}{\to }-\underset{b}{\to }$ , where $\underset{a}{\to }=\hat{i}+\hat{j}+\hat{k}$ and $\underset{b}{\to }=\hat{i}+2\hat{j}+3\hat{k}$

Answer 1

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

$\overrightarrow{b}=\hat{i}+2\hat{j}+3\hat{k}$

$\overrightarrow{a}+\overrightarrow{b}=2\hat{i}+3\hat{j}+4\hat{k}$

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

Now, let $\overrightarrow{c}$ be the vector perpendicular to the vectors $\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}-\overrightarrow{b}$.

$\overrightarrow{c}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 3& 4\\ 0& -1& -2\end{array}\right|=-2\hat{i}+4\hat{j}-2\hat{k}$

Now, $\hat{c}=\frac{-1}{\sqrt{6}}\hat{i}+\frac{2}{\sqrt{6}}\hat{j}-\frac{1}{\sqrt{6}}\hat{k}$

Therefore, required vector = $6\hat{c}=-\sqrt{6}\hat{i}+2\sqrt{6}\hat{j}-\sqrt{6}\hat{k}$.