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.
$\therefore \left|\begin{array}{ccc}a& b& c\\ 1& 1& 0\\ 0& 1& 1\end{array}\right|=0$
$\Rightarrow a-b+c=0$
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. $\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ a& b& c\\ 2& -2& -4\end{array}\right|=0$
$\Rightarrow \hat{i}\left(-4b+2c\right)-\hat{j}\left(-4a-2c\right)+\hat{k}\left(-2a-2b\right)=0$
$\Rightarrow -4b+2c=0$, $4a+2c=0$, $2a+2b=0$
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 $\hat{i}-\hat{j}-2\hat{k}$.