Orthogonality

Orthogonality generalises to an inner product space.

Mathematical definition

Let vector space $V$ be an inner product space.

Two vectors $u,v\in V$ are orthogonal if $\langle u, v \rangle=0$.

$u\bot v$ denotes this orthogonality.

Properties

$\vec 0$ is orthogonal to every vector

$$ \forall v \in V, \vec 0 \bot v $$

Proof:

$$ Recall the properties of a vector space by definition. By additive identity, $\vec 0 +\vec 0 = \vec 0$

Therefore,

$$ \langle \vec 0, u \rangle = \langle \vec 0+\vec 0, u\rangle $$

Recall the properties of the inner product by definition, by linearity in the 1st slot,

$$ \langle \vec 0+\vec 0, u\rangle = \langle\vec 0, u \rangle+\langle \vec 0, u\rangle $$

Subtracting $\langle \vec 0, u\rangle$ from both sides gives

$$ \langle \vec 0, u \rangle=0. $$