Question

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

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

Answer 1

The correct option is C $\hat{j}-\hat{k}$

Let $\overrightarrow{r}$ be coplanar with given two vectors
$\therefore \overrightarrow{r}=x(\hat{i}+\hat{j}+2\hat{k})+y(\hat{i}+2\hat{j}+\hat{k})$ ...(i)
Also, $\overrightarrow{r}$ is perpendicular to vector $(\hat{i}+\hat{j}+\hat{k})$
$\therefore \overrightarrow{r}.(\hat{i}+\hat{j}+\hat{k})=0$
$\Rightarrow (x+y),+(x+2y)+(2x+y)=0$
$\Rightarrow 4(x+y)=0$
$\Rightarrow y=-x$
$\therefore$ From (i), $\overrightarrow{r}=x(\hat{i}+\hat{j}+2\hat{k}-\hat{i}+2\hat{j}-\hat{k})$
$=x(-\hat{j}+\hat{k})$
Taking $x=\pm 1,$
$\overrightarrow{r}=-\hat{j}+\hat{k}\text{or}\overrightarrow{r}=\hat{j}-\hat{k}$