Symmetry of Orthogonality and the Pythagorean Identity

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, let $\lVert\cdot\rVert$ be the induced norm, let $0_{V}$ be the zero vector, and let $u,v\in V$. Then the following hold.

1. (Symmetry) $u$ and $v$ are orthogonal if and only if $v$ and $u$ are orthogonal.

2. (Pythagorean identity) If $u$ and $v$ are orthogonal, then

$$\lVert u+v\rVert^{2}=\lVert u\rVert^{2}+\lVert v\rVert^{2}.$$

3. (Zero vector) $0_{V}$ and $v$ are orthogonal.