Question

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

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

Answer 1

Answer: (C)

$\frac{3\widehat{i}\, -\, \widehat{k}\, }{\sqrt {10}}$

Answer 2

As we know that, a vector coplanar to $\overrightarrow {a} , \overrightarrow {b}$ and
orthogonal to $\overrightarrow {c}$ is $\lambda \\{(\overrightarrow {a} \times \overrightarrow {b}) \times \overrightarrow {c} \\}.$
$\therefore$ A vector coplanar to $(2\widehat{i} + \widehat{j} + \widehat{k}), (\widehat{i} + \widehat{j} + \widehat{k})$ and
orthogonal to $3\widehat{i} + 2\widehat{j} + 6\widehat{k}$
\hspace10mm $=\lambda \, [\\{(2\widehat{i} + \widehat{j} + \widehat{k}) \times (\widehat{i} + \widehat{j} + \widehat{k})\\} \, \, \times (3\widehat{i} + 2\widehat{j} + 6\widehat{k})]$
\hspace10mm $=\lambda \, [(2\widehat{i} + \widehat{j} + 3\widehat{k}) \times (3\widehat{i} + 2\widehat{j} + 6\widehat{k})] = \lambda( 21\widehat{j} - 7\widehat{k})$
$\therefore$ Unit vector $= +\frac{( 21\widehat{j} - 7\widehat{k})}{\sqrt {(21)^2+(7)^2}} \, = + \frac{(3\widehat{i} - \widehat{k})}{\sqrt 10}$