Linear dependent points on plane in $R^3$

Question

Let A, B, C, D be four distinct points in 3-dimensional space R3. What does it tell us about the four points if the three vectors AB,AC,AD are linearly dependent?

I understand that points must all lie on a single plane and that the three vectors since they are linear depend will lie in a sub space of 2-dimensions but the question is worth 7 Marks so I am stuck on what the answer would be

Answer 1

It tells that they lay in the same plane. To see it, let's discuss two cases:

Case one: $\vec{AB}$, $\vec{AC}$ are linear dependent, then they both lay on the same line, and with $\vec{AD}$ the share at least one plane.

Case two: $\vec{AB}$, $\vec{AC}$ are linear independent, then $\vec{AD}$ is dependent on them, which means it can be written as a linear combination of both of them which means, it lays on the plane formed be the vectors.

Answer 2

That the four points lie on the same plane, if the matrix formed by the three given vectors has rank 2, or on the line , if it has rank 1.