mulivariable calculus-distance and planes

Question

With 4 points A B C D, how do I find the distance from point D to the plane through A, B, C? This is a rather basic calc question I know but I'm not sure where to start. I imagine I'd probably have to find the vectors of a few of the points and then use some cross products and dot products

Answer 1

The vector $\vec{AB} \times\vec{AC}$ is normal to the plane through $A,B,C$. So the distance from $D$ to this plane is part of $\vec{AD}$ that is parallel to this vector or: $$d(D, ABC) = \frac{\vec{AD}\cdot(\vec{AB} \times\vec{AC})}{|(\vec{AB} \times\vec{AC})|}$$