Question

The vectors which is/are coplanar with vectors $\hat{i}+\hat{j}+2\hat{k}$ and $\hat{i}+2\hat{j}+\hat{k},$ are 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: (C), (D)

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

Explanation for the correct option:

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

Let the vector on the plane of $\overrightarrow{a}$ and $\overrightarrow{b}$ is;

$$\begin{gathered} \overrightarrow{r}=\overrightarrow{a}\lambda +\mu \overrightarrow{b} \\ =\lambda \left(\hat{i}+\hat{j}+2\hat{k}\right)+\mu \left(\hat{i}+2\hat{j}+\hat{k}\right) \\ =\left(\lambda +\mu \right)\hat{i}+\left(\lambda +2\mu \right)\hat{j}+\left(2\lambda +\mu \right)\hat{k} \end{gathered}$$

Also,

$$\begin{gathered} \therefore \overrightarrow{r}·\overrightarrow{c}=0 \\ \Rightarrow \left(\lambda +\mu \right)·1+\left(\lambda +2\mu \right)·1+(2\lambda +\mu )·1=0 \\ \Rightarrow 4\lambda +4\mu =0 \\ \Rightarrow \lambda +\mu =0 \end{gathered}$$

Hence,$\left[rab\right]=0$

Therefore the vectors $\hat{i}-\hat{j}$ and $-\hat{j}+\hat{k}$ satisfy the conditions.

Hence the correct option is $\mathrm{Option}\left(\mathrm{C}\right)$and $\mathrm{Option}\left(\mathrm{D}\right)$