You ask only if it represents a plane, but I think you can say more. It actually represents the plane through $\displaystyle (x_1,y_1,z_1)$, $\displaystyle (x_2,y_2,z_2)$, and $\displaystyle (x_3,y_3,z_3)$, doesn't it?
- Hollywood
Yes, the last column must be all the same non-zero value if you want the plane to pass through each of the points (x1,y1,z1), etc. So you might as well make this common value 1.
To see why you get a plane, expand the determinant along the first row and you get ax +by +cz + d =0, where a, b, c and d are cofactors. Now to see (x1,y1,z1) satisfies this equation, replace the first row x y z 1 by x1 y1 z1 1, and you have a determinant with two rows the same, so the determinant is 0. Notice though if the three points are collinear, you don't get a plane at all; the determinant is identically 0!