Let $V$ be an inner product space. Prove $\langle x, 0\rangle = \langle 0, x\rangle = 0$ for $x \in V$.

Question

Let $V$ be an inner product space. For $x \in V$, prove $\langle x, 0\rangle = \langle 0, x\rangle = 0$.

Answer 1

Here's a hint: $\langle x,0\rangle = \langle x,2\cdot 0\rangle$.

Answer 2

A different hint: $\langle x,0 \rangle = \langle x, x + (-x) \rangle$.

Answer 3

PROOF: We want to show that $\langle x, 0 \rangle = \langle 0, x\rangle = 0$ for any $x \in V$. Observe that $$\begin{align}\langle x,0\rangle & = \langle x, x + (-x) \rangle\\&=\langle x,x\rangle + \langle x,-x\rangle \\ &=\langle x, x \rangle - \langle x, x\rangle \tag{Factor out $-1$}\\&=0.\end{align}$$ Likewise, $$\begin{align}\langle 0,x\rangle & = \langle x + (-x), x \rangle\\&=\langle x,x\rangle + \langle -x,x\rangle \\ &=\langle x, x \rangle - \langle x, x\rangle \tag{Factor out $-1$}\\&=0.\end{align}$$