How to use Vectors to find the centroid of a tetrahedron?

Question

Suppose that four points - A,B,C,D (with position vectors a,b,c,d) are the vertices of a tetrahedron. And the mid points of BC, CA, AB, AD, BD, CD are denoted by P,Q,R,U,V,W.

Using these info, I tried finding the centroid by finding the mid-point of vectors PU, QV and RW. But it isn't working out for me.

Can someone please help me with this?

Answer 1

The centroid is $(a+b+c+d)/4$. You don't need the midpoints of the sides.

You might understand this better if you think about the centroid of a triangle. It lies $2/3$ of the way along the median from each vertex.

Answer 2

Also, you can take: $\frac{P+U}{2}$ or $\frac{Q+V}{2}$ or $\frac{R+W}{2}.$