Question

The unit vector which is orthogonal to the vector $3\hat{i}+2\hat{j}+6\hat{k}$ and is coplanar with the vectors $2\hat{i}+\hat{j}+\hat{k}and\hat{i}-\hat{j}+\hat{k}$

• A. $\frac{2\hat{i}-6\hat{j}+\hat{k}}{\sqrt{41}}$
• B. $\frac{2\hat{i}-3\hat{j}}{\sqrt{13}}$
• C. $\frac{3\hat{j}-\hat{k}}{\sqrt{10}}$
• D. $\frac{4\hat{i}+3\hat{j}-3\hat{k}}{\sqrt{34}}$

Answer 1

The correct option is C $\frac{3\hat{j}-\hat{k}}{\sqrt{10}}$

As we know that, a vector coplanar to $\overrightarrow{a},\overrightarrow{b}$ and orthogonal to $\overrightarrow{c}$ is $\lambda (\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}.$
$\therefore \text{A vector coplanar to}(2\hat{i}+\hat{j}+\hat{k}),(\hat{i}-\hat{j}+\hat{k})\text{and orthogonal to}3\hat{i}+2\hat{j}+6\hat{k}=\lambda [\left\{\right(2\hat{i}+\hat{j}+\hat{k})\times (\hat{i}-\hat{j}+\hat{k}\left)\right\}\times (3\hat{i}+2\hat{j}+6\hat{k}\left)\right]=\lambda [(2\hat{i}-\hat{j}-3\hat{k})\times (3\hat{i}+2\hat{j}+6\hat{k})]=\lambda (21\hat{j}-7\hat{k})\therefore Unitvector=+\frac{(21\hat{j}-7\hat{k})}{\sqrt{(21{)}^2+(7{)}^2}}=+\frac{(3\overrightarrow{j}-\hat{k})}{\sqrt{10}}$