Question

Let $A,B,C$ be three points with position vectors $\hat{i}+\hat{j},\hat{i}-\hat{j}$ and $l\hat{i}+m\hat{j}+n\hat{k}$ respectively and $O$ be the origin, then $\overrightarrow{OA},\overrightarrow{OB},\overrightarrow{OC}$ are coplanar if

• A. $l=m=n=1$
• B. $l=m=n=0$
• C. $l=1$, $m$ & $n$ are arbitrary scalars
• D. $l$ and $m$ are arbitrary scalars $n=0$

Answer 1

Answer: (D)

$l$ and $m$ are arbitrary scalars $n=0$

Equation of coplanarity is given by

$l\hat{i}+m\hat{j}+n\hat{k}=\mu (\hat{i}+\hat{j})+\lambda (\hat{i}-\hat{j})$
$L=\mu +\lambda$
$m=\mu -\lambda$
$n=0$ $\mu =\frac{L+m}{2}\lambda =\frac{L-m}{2}$
So $l,m$ be any value where $n=0$