The cross product of two vectors will be perpendicular to the plane containing both vectors. And that means that the dot product of that perpendicular with any vector in the plane will be 0. That is, .
To form vectors from one point to another, subtract the coordinates. That will give the three vectors forming the three sides of the triangle with the points as vertices.
The "triple product" above can be easily calculated by finding the 3 by 3 determinant having the vectors as rows. The points are co-planar if and only if that determinant is 0.