Prove relationship regarding the scalar product

Question

For 2 vectors $a,b$ $\in \mathbb{R}^n$ and all entries in the vectors are $\geq 1$ is the following relationship true ? :

$\langle a,b \rangle$ $ \leq$ $0.5 \langle a,a \rangle + 0.5 \langle b,b \rangle$

If yes is there a simple way to prove this or a geometric intuition behind this relationship ?

Answer 1

You know that $$\|a-b\|^2=\langle a-b,a-b \rangle \ge 0.$$

Can you simplify the expression $\langle a-b,a-b \rangle$ in some way? Can you get the inequality you are asking about from this?

$$\langle a-b,a-b \rangle = \langle a,a \rangle - 2\langle a,b \rangle + \langle b,b \rangle \ge 0$$
$$\langle a,a \rangle + \langle b,b \rangle \ge 2\langle a,b \rangle $$

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You may also have a look how this relates to law of cosines.