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 $2\hat{i}+\hat{j}+\hat{k}\text{and}\hat{i}-\hat{j}+\hat{k}$ is:

• A. $\frac{2\hat{i}-3\hat{j}}{\sqrt{13}}$
• B. $\frac{2\hat{i}-6\hat{j}+\hat{k}}{\sqrt{41}}$ ​​​​​​​
• 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, a vector coplanar to $a,b$ and orthogonal to $c$ is $\lambda \left\{\right(a\times b)\times c\}$ $\therefore$ A vector coplanar to $(2\hat{i}+\hat{j}+\hat{k}),(\hat{i}-\hat{j}+\hat{k})$ and orthogonal to $(3\hat{i}+2\hat{j}+6\hat{k})$. $=\lambda \left[\right\{2\hat{i}+\hat{j}+\hat{k})\times (\hat{i}-\hat{j}+\hat{k}\left)\right\}\times (3\hat{i}+2\hat{j}+6\hat{k})]$ $=\lambda (-21\hat{j}+7\hat{k})$ $\therefore \text{A unit vector is}\pm \frac{(a\times b)\times c}{(a\times b)\times c}$ $=\pm \frac{-21\hat{j}+7\hat{k}}{\sqrt{{21}^2+7^2}}=\frac{\pm (3\hat{j}-\hat{k})}{\sqrt{10}}$