Question

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

• A. $R$
• B. $R-\left\{1\right\}$
• C. $R-\left\{2\right\}$
• D. They are always non-coplanar $\forall mϵR$

Answer 1

Answer: (D)

They are always non-coplanar $\forall mϵR$

By the coplanarity condition, we have

$\left|\begin{array}{ccc}1& 1& m\\ 1& 1& m+1\\ 1& -1& m\end{array}\right|=0$

$\Rightarrow 1(m+m+1)-1(m-m-1)+m(-1-1)=0$
$\Rightarrow 2m+1+1-2m=0$
$\Rightarrow 2=0$
This is not possible and hence the vectors given are always non coplanar.