We know that cosθ = 0 when θ = 90°. From our formula for angle, we can see that cosθ = 0 if and only if u • v = 0. This gives us the following definition for orthogonality.
Two vectors u and v are considered to be orthogonal if and only if u • v = 0. This is denoted by,
u ⊥ v ⇔ u • v = 0
The symbol '⊥' denotes orthogonality. In R2 and R3, orthogonal vectors are equivalent to perpendicular vectors (remember that perpendicular lines or vectors are at a 90° right angle to one another.) In Rn, the definition of orthogonality allows us to generalize the idea of perpendicular vectors where our usual ideas of geometry don't always apply. The following example makes use of our definition of orthogonal vectors.
We can use orthogonality to state Pythagoras' Theorem in terms of vector lengths. Given two vectors u and v, we can say they are orthogonal if and only if ||u + v||2 = ||u||2 + ||v||2, that is
u ⊥ v ⇔ ||u + v||2 = ||u||2 + ||v||2