In a three-dimensional space, consider four points $P_0, P_1, P_2, and P$ that are coplanar. Let these points determine four vectors, respectively $X_0, X_1, X_2, X$. Visually, these vectors go from the origin to each of the four points.
In Elementary Matrix Algebra, 3rd ed. by Franz E. Hohn, in section 3.8 Hohn asserts that because the four points are coplanar, the vectors $X_1-X_0$, $X_2-X_0$, and $X-X_0$ must be coplanar vectors. I don't understand why this must be so. (Either an informal explanation or a rigorous justification could be helpful.)
I've looked through earlier sections of the book for principles that could be used to justify the claim, but I haven't been able to figure out a justification for it. (There's a figure (3.12) that illustrates the idea. This contains representations of the three difference vectors, and they do appear to be coplanar, but I'm having trouble perceiving the angles in the flattened 3-D representation well enough to see how these vectors would result from the addition of each of the three original vectors with $-X_0$.)
Think of the vector $X_1 - X_0$ as the vector $P_0P_1$, and do the same with the other three vectors you're given. Since the points $P_0, P_1, P_2, P$ lie in a plane, your three vectors are actually vectors within this plane, and so must be coplanar.
Another way to intuitively understand this is that taking every point in the plane $P_0P_1P_2P$ and subtracting $X_0$ from it translates the entire plane, sending $P_0$ to the origin. Now any three vectors in this plane are clearly coplanar.