Question

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-coplanar vectors and $\lambda$ is a real number, then the vectors $\overrightarrow{a}+2\overrightarrow{b}+3\overrightarrow{c},\lambda \overrightarrow{b}+4\overrightarrow{c}$ and $(2\lambda -1)\overrightarrow{c}$ are non-coplanar for

• A. All values of $\lambda$
• B. All except one value of $\lambda$
• C. All except two values of $\lambda$
• D. No value of $\lambda$

Answer 1

Answer: (C)

All except two values of $\lambda$

The three vectors $(\overrightarrow{a}+2\overrightarrow{b}+3\overrightarrow{c}),(\lambda \overrightarrow{b}+4\overrightarrow{c})$ and $(2\lambda -1)\overrightarrow{c}$ are coplanar if

$\left|\begin{array}{ccc}1& 2& 3\\ 0& \lambda & 4\\ 0& 0& 2\lambda -1\end{array}\right|=0$

$\Rightarrow (2\lambda -1)\left(\lambda \right)=0\Rightarrow \lambda =0,1/2$

Therefore these three vectors are non-coplanar for all expect two values of $\lambda$ $($ i.e.$0,1/2)$