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. For $\lambda ϵR$
• B. For $\lambda \ne \frac{1}{2}$
• C. For $\lambda \ne 0,\frac{1}{2}$
• D. For $\lambda ϵ\varphi$

Answer 1

The correct option is C For $\lambda \ne 0,\frac{1}{2}$

Condition for coplanarity

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

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

$\Rightarrow \lambda =0,\frac{1}{2}$