Browse Questions

# If the vectors $\overrightarrow a+\lambda \overrightarrow b+3\overrightarrow c,\:-2\overrightarrow a+3\overrightarrow b-4\overrightarrow c\:and\:\overrightarrow a-3\overrightarrow b+5\overrightarrow c$ are coplanar and $\overrightarrow a,\overrightarrow b,\overrightarrow c$ are non coplanar, then $\lambda=?$

Given that $\overrightarrow x=\overrightarrow a+\lambda \overrightarrow b+3\overrightarrow c,\:\overrightarrow y=-2\overrightarrow a+3\overrightarrow b-4\overrightarrow c\:\:and\:\:\overrightarrow z=\overrightarrow a-3\overrightarrow b+5\overrightarrow c$ are coplanar.
$\Rightarrow\:[\overrightarrow x\:\overrightarrow y\:\overrightarrow z]=0$
$\Rightarrow\:\left |\begin {array}{ccc} 1 & \lambda & 3\\ -2 & 3 & -4 \\ 1 & -3 & 5\end {array}\right |.[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]=0$
$\Rightarrow\:\big[(15-12)-\lambda(-10+4)+3(6-3)\big]. [\overrightarrow a\:\overrightarrow b\:\overrightarrow c]=0$
$\Rightarrow\:(6\lambda+12) [\overrightarrow a\:\overrightarrow b\:\overrightarrow c]=0$
But it is also given that $\overrightarrow a,\overrightarrow b,\overrightarrow c$ are non coplanar.
$\therefore [\overrightarrow a\:\overrightarrow b\:\overrightarrow c] \neq 0$
$\therefore\:6\lambda+12=0$ or $\lambda=-2$