Question

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

• A. all values of $\lambda$
• B. all except one value of $\lambda$
• C. all except two value of $\lambda$
• D. None value of $\lambda$

Answer 1

Answer: (C)

all except two value of $\lambda$

$A=\overrightarrow{a}+\overline{2b}+\overrightarrow{3c},\overleftarrow{B}=\lambda b+4c,\overrightarrow{C}=\left(2\lambda -1\right)\overrightarrow{c}are$

$non-coplanarif$

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

$1\left(2{\lambda }^2-\lambda \right)\ne 0$

$\lambda \left(2\lambda -1\right)\ne 0$

$sothisispossibleif\lambda \ne 0\&\lambda \ne \frac{1}{2}$

$soexcepttwovalueof\lambda$