Question

The set of values of $m$ for which the vectors $\overrightarrow{i}+\overrightarrow{j}+m\overrightarrow{k}$, $\overrightarrow{i}+\overrightarrow{j}+(m+1)\overrightarrow{k}$, $\overrightarrow{i}-\overrightarrow{j}+m\overrightarrow{k}$ are non coplanar is

• A. $R$
• B. $R-\left\{1\right\}$
• C. $R-\{-2\}$
• D. $\varnothing$

Answer 1

The correct option is A $R$

$\hat{i}+\hat{j}+m\hat{k}=\lambda (\hat{i}+\hat{j}+(m+1\left)\hat{k}\right)+m(\hat{i}-\hat{j}+m\hat{k})$
$\lambda +m=1$......(1)
$\lambda -m=1$......(2)
$\lambda =1$ and $m=0$
$m=(\lambda +m)m+\lambda$......(3)
$0=\lambda$
So they can't be coplanar for any value of $m$