If a= i-2 j+3 k, b=2 i+3 j- k and c=r i+ j+(2r-1) k are three vectors such that c is paral…

Question

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

Accepted Answer

$\vec c$ parallel to the plane of $\vec a,\vec b$ means $\vec a,\vec b,\vec c$ are coplanar:

$$\begin{vmatrix}1&-2&3\2&3&-1\r&1&2r-1\end{vmatrix}=0 \implies (6r-2)+2(5r-2)+(6-9r)=0 \implies 7r=0 \implies r=0$$

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