Problem finding the plane passing through 3 points

Question

What if solving this problem...

...I take the 3 vectors difference in this graph

By imposing that parallelepiped volume is zero, I would get

And so

But this, full of variables squared and cubed doesn't look at all a plane! What went wrong?

Answer 1

Nothing went wrong, just use elementary row operations ! Subtract the first row from the second and third, to recover the form stated in the book \begin{eqnarray*} \begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ x-x_2 & y-y_2 & z-z_2 \\ x-x_3 & y-y_3 & z-z_3 \\ \end{vmatrix} = \begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ x_1-x_2 & y_1-y_2 & z_1-z_2 \\ x_1-x_3 & y_1-y_3 & z_1-z_3 \\ \end{vmatrix} =0. \end{eqnarray*}