Question

A vector of magnitude $5\sqrt{5}$ coplanar with vectors $\hat{i}+2\hat{j}$ & $\hat{j}+2\hat{k}$ and the perpendicular vector $2\hat{i}+\hat{j}+2\hat{k}$ is

• A. $\pm 5(5\hat{i}+6\hat{j}-8\hat{k})$
• B. $\pm \sqrt{5}(5\hat{i}+6\hat{j}-8\hat{k})$
• C. $\pm 5\sqrt{5}(5\hat{i}+6\hat{j}-8\hat{k})$
• D. $\pm (5\hat{i}+6\hat{j}-8\hat{k})$

Answer 1

The correct option is D $\pm (5\hat{i}+6\hat{j}-8\hat{k})$

Out of the four options, only one option has a magnitude of $5\sqrt{5}.$
Still, let the vector required be $p\hat{i}+q\hat{j}+r\hat{k}.$

The coplanarity condition gives $\left|\begin{array}{ccc}1& 2& 0\\ 0& 1& 2\\ p& q& r\end{array}\right|=0$

$\Rightarrow 1(r-2q)-2(0-2p)+0=0$
$\Rightarrow r-2q+4p=0$

Its perpendicular condition gives $2p+q+2r=0$
Solving these, we have $5q=6p$ and thus we get option $D$ as correct.