# Dot product and orthogonality?

Please only use the following definition of the dot product:

$u \dot{} v = u^{T}v$

Using the above definiton only (not the cosine definition) why would the dot product being zero imply the angle between the vectors is 90/-270?

This isn't homework. Please avoid use of the Pythagorean theorem, since my book uses the above result to prove it.

• Do you mean $\pm\pi/2$? – Olivier Bégassat Dec 3 '13 at 2:22
• @OlivierBégassat Yes I do. Sorry about that. – dfg Dec 3 '13 at 2:34

By assumption, we have $$||u+v||^2=(u+v)^T(u+v)=||u||^2+2u\cdot v+||v||^2=||u||^2+||v||^2$$ And then the orthogonality results from Pythagorean theorem.