Question

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-coplanar vector 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 when

• A. $\mu ϵR$
• B. $\lambda =\frac{1}{2}$
• C. $\lambda =0$
• D. All value are correct

Answer 1

Answer: (D)

All value are correct

For coplanar condition

$\begin{array}{c}\left(\overrightarrow{A}\times \overrightarrow{B}\right)\cdot \overrightarrow{C}=0\\ \left[\left(\overrightarrow{a}+2\overrightarrow{b}+3\overrightarrow{c}\right)\times \left(\lambda \overrightarrow{b}+\mu \overrightarrow{c}\right)\right]\cdot \overrightarrow{c}=0\\ \lambda \left(2\lambda -1\right)=0\\ \lambda =0,\lambda =\frac{1}{2}and\mu \in R\end{array}$