Statement-1: For any three vectors $\vec a, \vec b, \vec c$

$\begin{bmatrix}\vec a×\vec b&\vec b×\vec c&\vec c×\vec a\end{bmatrix}=0$

Statement-2: If $\vec p,\vec q,\vec r$ are linearly dependent vectors then they are coplanar.

Statement-1 is False, Statement-2 is True.

If $\vec p,\vec q,\vec r$ are linearly independent vectors, then there exist scalars x, y, z not all zero such that

$x\vec p+y\vec q+z\vec r = \vec 0$

$⇒\vec p=(-\frac{y}{x})\vec q+(\frac{-z}{x})\vec r$

$⇒\vec p,\vec q,\vec r$ are coplanar.

So, statement-2 is true.

We know that $\begin{bmatrix}\vec a×\vec b&\vec b×\vec c&\vec c×\vec a\end{bmatrix}=[\vec a\,\,\vec b\,\,\vec c]^2 ≠0$ unless $\vec a, \vec b, \vec c$ are coplanar.

So, statement-2 is not true.