# Thread: Tetrahedron and the dot product

1. ## Tetrahedron and the dot product

Here is the question
"Two pairs of opposite edges of a tetrahedron are perpendicular. Prove that the third pair is also perpendicular."

I let the 3 sides be a, b and c
assume a is perpendicular to b
assume b and c are perpendicular

then a.b=0 and b.c=o

need to show a.c=o

but I am not sure how to proceed.

Cheers
Cabouli

2. Originally Posted by Cabouli
Here is the question
"Two pairs of opposite edges of a tetrahedron are perpendicular. Prove that the third pair is also perpendicular."

I let the 3 sides be a, b and c
assume a is perpendicular to b
assume b and c are perpendicular

then a.b=0 and b.c=o

need to show a.c=o

but I am not sure how to proceed.
Suppose the tetrahedron is OABC, with one vertex at the origin, and the others at points given by the vectors $\displaystyle a,\ b,\ c$. You are told that $\displaystyle a.(c-b) = b.(a-c) = 0$, and you want to prove that $\displaystyle c.(b-a) = 0$. Just use linearity: $\displaystyle a.(c-b) = a.c-a.b,\ldots$.