Let $\vec a = 2\hat i +3\hat j-\hat k$ and $\vec b =\hat i-2\hat j+3\hat k$ Then, the value of λ for which the vector $\vec c=2\hat i+\hat j+(2λ-1)\hat k$ is parallel to the plane containing $\vec a$ and $\vec b$, is

0

We are given that $\vec c$ is parallel to the plane containing $\vec a$ and $\vec b$

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

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