Orthogonal in inner product space

Question

Let $(X,<.>)$ is an inner product space prove that $x$ and $y$ are orthogonal if and only if $||x+αy|| \ge ||x||$ for any scalar $α$ . The first direction if $x$ and $y$ are orthogonal then$||x+αy||^2=||x||^2+|α|^2||y||^2 \ge ||x||^2 $ so $||x+αy|| \ge ||x||$ . But the second direction if $||x+αy|| \ge ||x||$ for any scalar $α$ Implies $x$ and $y$ are orthogonal how can I prove that ?

Answer 1

$$\langle x+\alpha y,x+\alpha y\rangle\geq\langle x,x\rangle\forall \alpha\\ \langle x,x\rangle+2\alpha\langle x,y\rangle+\alpha^2\langle y,y\rangle\geq\langle x,x\rangle\forall\alpha\\ 2\alpha \langle x,y\rangle+\alpha^2\langle y,y\rangle \geq0\forall\alpha$$ But the left-hand side equals zero if $\alpha=0$ and if $\alpha=-2\langle x,y\rangle/\langle y,y\rangle$. If $\langle x,y\rangle\neq0$ then this is a quadratic in $\alpha$ which has two zeros, and will be negative for some $\alpha$. In particular, $\alpha=-<x,y>/<y,y>$ makes the left-hand side equal to $-<x,y>^2/<y,y>$