Determine if the triangle is right-angled using vectors.

Question

Use vectors to decide whether the triangle with vertices $P\langle 1, -3, -2\rangle, Q\langle2, 0, -4\rangle$, and $R\langle6, -2, -5\rangle$ is right-angled.

I tried taking the dot product $P\cdot Q$, $Q \cdot R$, and $P \cdot R$ but none are zero. However the triangle is right-angled. Any suggestions?

Answer 1

The direction vectors for the three sides are given by the differences in the vertices, not the vertices themselves. So consider the vectors

$$P - Q, P - R, Q - R$$ and see whether any of these are orthogonal.

Answer 2

The three sides of the triangle correspond to vectors $P-Q$, $Q-R$, $R-P$. Take dot products of those.