Question

The vector(s) which is/are coplanar with vectors $\hat{i}+\hat{j}+2\hat{k}$ and $\hat{i}+2\hat{j}+\hat{k}$, and perpendicular to the vector $\hat{i}+\hat{j}+\hat{k}$ is/are

• A. $\hat{j}-\hat{k}$
• B. $-\hat{i}+\hat{j}$
• C. $\hat{i}-\hat{j}$
• D. $-\hat{j}+\hat{k}$

Answer 1

Answer: (A), (D)

$\hat{j}-\hat{k}$
$-\hat{j}+\hat{k}$

Let $\overrightarrow{a}=\hat{i}+\hat{j}+\widehat{2k}$
$\overrightarrow{b}=\hat{i}+\widehat{2j}+\hat{k}$

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

Required vector, $\overrightarrow{v}=\lambda [\overrightarrow{c}\times (\overrightarrow{a}\times \overrightarrow{b}\left)\right]$

$=\lambda \left[\right(\overrightarrow{c}.\overrightarrow{b})\overrightarrow{a}-(\overrightarrow{c}.\overrightarrow{a}\left)\overrightarrow{b}\right)]$
$=\lambda [-4\hat{j}+4\hat{k}]$

So,

$\overrightarrow{v}=-\hat{j}+\hat{k}$

or

$\overrightarrow{v}=\hat{j}-\hat{k}$