Question

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

• A. All values of $\mu$
• B. $\lambda =\frac{1}{2}$
• C. $\lambda =0$
• D. No value of $\lambda$

Answer 1

Answer: (B), (C)

The correct options are
B $\lambda =\frac{1}{2}$
C $\lambda =0$

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

$(2\lambda -1)(\lambda -0)=0$

Or, $\lambda (2\lambda -1)=0$

$\therefore either\lambda =0$ or $2\lambda -1=0\Rightarrow \lambda =1/2$