If $\vec a =\hat i-2\hat j+3\hat k, \vec b = 2\hat i+\hat j+(2r-1)\hat k$ are three vectors such that $\vec c$ is parallel to the plane of $\vec a$ and $\vec b$, then r is equal to:

0

It is given that $\vec c$ is parallel to the plane of $\vec a$ and $\vec b$

$∴\vec c⊥\vec a×\vec b$

$⇒\vec c.(\vec a×\vec b)=0⇒ [\vec a\,\, \vec b\,\, \vec c]=0$

$⇒\begin{vmatrix}1&-2&3\\2&3&-1\\r&1&2r-1\end{vmatrix}=0$

$⇒(6r-2)+2 (5r-2)+(6-9r)=0⇒7r=0 ⇒r = 0$