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# If $\overrightarrow a=2\hat i+3\hat j-\hat k\:\:and\:\:\overrightarrow b=\hat i-2\hat j+3\hat k$, then the value of $\lambda$ for which $\overrightarrow c=\lambda\hat i+\hat j+(2\lambda+1)\hat k$ is parallel to the plane containing $\overrightarrow a\:\:and\:\:\overrightarrow b$ is

Given that $\overrightarrow c$ is parallel to the plane containing $\overrightarrow a\:and\:\overrightarrow b$
$\Rightarrow \overrightarrow c\:\:is\:\:\perp$ to $\overrightarrow a\times\overrightarrow b$
$\Rightarrow\:\overrightarrow c.(\overrightarrow a\times\overrightarrow b)=0$
$\Rightarrow\:\left|\begin {array}{ccc}2 & 3 &-1\\1 &-2 &3\\\lambda & 1 & 2\lambda -1\end {array}\right|=0$
$\Rightarrow\:\lambda=0$