Given four three-dimensional points $a=(a_x,a_y,a_z),b=(b_x,b_y,b_z),c=(c_x,c_y,c_z),d=(d_x,d_y,d_z)$, the determinat is defined as:
\begin{equation}\begin{bmatrix}a,b,c,d\end{bmatrix} := \begin{vmatrix}a_x&a_y&a_z&1\\b_x&b_y&b_z&1\\c_x&c_y&c_z&1\\d_x&d_y&d_z&1\end{vmatrix} \end{equation}
\begin{equation} = \begin{vmatrix}a_x-d_x&a_y-d_y&a_z-d_z\\b_x-d_x&b_y-d_y&b_z-d_z\\c_x-d_x&c_y-d_y&c_z-d_z\end{vmatrix} \end{equation}
The sign of the determinant tells whether the point d is above, on, or below a plane through a, b, and c.
The above is from a paper I'm trying to understand. I'm not familiar with the theorem used, and my searches have come up empty. Why does this determinant give the position of d relative to the abc plane? Why not the position of c relative to the abd plane?
Also I see that the first determinant is equal to the second one, but how do you get from the first to the second? Am I missing another theorem, or has the author simply (cleverly) spotted this correlation?
This is probably clearer if you do some simple row operations on your matrix to write it in the form
$$ \det \begin{bmatrix} a_x & a_y & a_z & 1 \\ b_x - a_x & b_y - a_y & b_z - a_z & 0 \\ c_x - a_x & c_y - a_y & c_z - a_z & 0 \\ d_x - a_x & d_y - a_y & d_z - a_z & 0 \end{bmatrix} $$
Now, expanding the determinant along the fourth column just gives
$$ -\det \begin{bmatrix} b_x - a_x & b_y - a_y & b_z - a_z \\ c_x - a_x & c_y - a_y & c_z - a_z \\ d_x - a_x & d_y - a_y & d_z - a_z \end{bmatrix} = -\det (\vec{AB}, \vec{AC}, \vec{AD})$$
The sign of this determinant tells you whether the vectors $ \vec{AB} $, $ \vec{AC} $ and $ \vec{AD} $ are positively or negatively oriented in three dimensional space. If the determinant is zero then $ \vec{AD} $ is a linear combination of $ \vec{AB} $ and $ \vec{AC} $, which means $ D $ lies on the plane containing $ A, B $ and $ C $. Otherwise the sign tells you which side of the plane $ D $ is on, though keep in mind this depends explicitly on the ordering of the points $ B $ and $ C $ (swapping them changes the sign of the determinant) so it isn't an intrinsic property of the plane itself, but rather dependent on the choice and order of points used to generate it.