Equation of a plane in vector form

Equation of a plane in vector form

I have a basic doubt. If we are given 3 points in space then the equation
of the plane passing through them is given by:
$$ \Bigg[\vec{r} - \vec{a}, \vec{a} - \vec{b}, \vec{a} - \vec{c}\Bigg] = 0$$
where $ \vec{r} $ is the position vector of the variable point and
$\vec{a}, \vec{b}, \vec{c}$ are the fixed points. This equation is true
because the box product of coplanar vectors is $0$. However, if we see
then even this:
$$ \Bigg[\vec{r} - \vec{a}, \vec{r} - \vec{b}, \vec{r} - \vec{c}\Bigg] = 0$$
satisfies the condition - the box product of 3 coplanar vectors is zero.
However, if we right this is Cartesian form, then we don't get a linear
equation and that wouldn't represent a plane, would it?