Proving a strange vector inequality in the euclidean space

Question

It seems to hold the following inequality in an euclidean reference frame $(x,y,z)$: $$\overrightarrow{U}\cdot\overrightarrow{U}\ge\sqrt{2}\left(\Omega_x+\Omega_y\right)$$ where: $$\overrightarrow{U}\equiv(a,b,c), (a,b,c\gt0)$$ $$\overrightarrow{\Omega}=\overrightarrow{V}\times\overrightarrow{W}$$ and: $$\overrightarrow{V}\equiv(0,a,b),\overrightarrow{W}\equiv(0,-c,b)$$ Is there any way to give a proof of that? Thanks.

Answer 1

I think you are asking if it is true that $$a^2+b^2+c^2\geq \sqrt{2}(ab+bc).$$

Well, \begin{align*} a^2+b^2+c^2-\sqrt{2}(ab+bc) & =\left(a-\frac{b}{\sqrt{2}}\right)^2 + \left(c-\frac{b}{\sqrt{2}}\right)^2 \geq 0. \end{align*}

By the way, notice that your inequality is true for any $a,b,c\in\mathbb{R}$.