Question

Vector coplanar with vectors $\hat{i}+\hat{j}$ and $\hat{j}+\hat{k}$ and parallel to the vector $2\hat{i}-2\hat{j}-4\hat{k}$, is

• A. $\hat{i}-\hat{k}$
• B. $\hat{i}-\hat{j}-2\hat{k}$
• C. $\hat{i}+\hat{j}-\hat{k}$
• D. $3\hat{i}+3\hat{j}-6\hat{k}$

Answer 1

Answer: (B)

Let vector be $a\hat{i}+b\hat{j}+c\hat{k}$.
$\because a \hat{i}+b\hat{j}+c\hat{k}$, $\hat{i}+\hat{j}$ and $\hat{j}+\hat{k}$ are coplanar.
$\therefore \begin{vmatrix}a&b&c\\ 1&1&0\\ 0&1&1\end{vmatrix} = 0$
$\Rightarrow a - b + c = 0$
Also, $\left(a\hat{i}+b\hat{j}+c\hat{k}\right)$ is parallel to $\left(2\hat{i}-2\hat{j}-4\hat{k}\right)$
$\therefore \left(a\hat{i}+b\hat{j}+c\hat{k}\right)\times \left(2\hat{i}-2\hat{j}-4\hat{k}\right) = \vec{0}$
i.e. $\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ a&b&c\\ 2&-2&-4\end{vmatrix} = 0$
$\Rightarrow \hat{i}\left(-4b+2c\right)-\hat{j}\left(-4a-2c\right)+\hat{k}\left(-2a-2b\right) = 0$
$\Rightarrow -4b + 2c = 0$, $4a + 2c = 0$, $2a + 2b =0$
i.e. $\frac{a}{-1} = \frac{b}{1} = \frac{c}{2}$ or
$\frac{a}{1} = \frac{b}{-1} = \frac{c}{-2}$
$\therefore $ Required vector is $\hat{i}-\hat{j} - 2\hat{k}$.