Question

Let $\overline{a}=2\overline{i}+\overline{j}+\overline{k},\overline{b}=\overline{i}+2\overline{j}-\overline{k}$ and a unit vector $\overline{c}$ be coplanar. If $\overline{c}$ is perpendicular to $\overline{a}$, then $\overline{c}$ is

Answer 1

$\overrightarrow{c}$ is coplanar with $\overrightarrow{a}$ &$\overrightarrow{b}$ and perpendicular to $\overrightarrow{a}$

$\therefore \overrightarrow{c}=\overrightarrow{a}\times (\overrightarrow{a}\times \overrightarrow{b})=(\overrightarrow{a}.\overrightarrow{b})\overrightarrow{a}-(\overrightarrow{a}.\overrightarrow{a})\overrightarrow{b}=\left(\right(2\hat{i}+\hat{j}+\hat{k}).(\hat{i}+2\hat{j}-\hat{k}\left)\right)(2\hat{i}+\hat{j}+\hat{k})=\left(\right(2\hat{i}+\hat{j}+\hat{k}).(2\hat{i}+\hat{j}+\hat{k}\left)\right)(\hat{i}+2\hat{j}-\hat{k})=(2+2-1)(2\hat{i}+\hat{j}+\hat{k})=(2+1+1)(\hat{i}+2\hat{j}-\hat{k})=3(2\hat{i}+\hat{j}+\hat{k})-4(\hat{i}+2\hat{j}-\hat{k})=6\hat{i}+3\hat{i}+3\hat{k}-4\hat{i}-8\hat{j}+4\hat{k}=2\hat{i}-5\hat{j}+7\hat{k}$