Prove an Inner Product is one

Question

Rather simple one, but I must be losing my mind.
Let $\vec v \in \mathbb{R} ^2 $

Define the inner product $<\vec v, \vec z>=4v_1z_1 + v_1z_2 + v_2z_1 + 4z_2z_2$

I have to prove that this is an inner product. Most of the "showings" are easy, but I can't show that this is positive if $\vec v \neq \vec 0$.

I set it up, but I got nothing.

$<\vec v, \vec v>=4v_1^2+2v_1v_2+4v_2^2 \gt 0$

What do I do with the middle term??

Answer 1

From the definition of the inner product,

$<\vec{v},\vec{v}> = 4v_1 ^2 + 2v_1 v_2 + 4v_2^2 = 3(v_1^2 + v_2^2) + v_1^2 +2v_1 v_2 +v_2^2 = 3(v_1^2 + v_2^2)+ (v_1 + v_2)^2 \geq 0$

Answer 2

$$4v_1^2+2v_1v_2+4v_2^2=3(v_1^2+v_2^2)+(v_1+v_2)^2 \geq 3(v_1^2+v_2^2)$$