1. ## Simple expression manipulation.

I'm only concerned about the expressions highlighted in red.

How did they get from the first to the second? I would have expected something like, $\displaystyle 2 (\vec{u} \cdot \vec{v}) \lVert \vec{u} \rVert \lVert \vec{v} \rVert$. Is that not correct? Or are they the same expression?

2. Originally Posted by scorpion007
I'm only concerned about the expressions highlighted in red.

How did they get from the first to the second? I would have expected something like, $\displaystyle 2 (\vec{u} \cdot \vec{v}) \lVert \vec{u} \rVert \lVert \vec{v} \rVert$. Is that not correct? Or are they the same expression?
I'll start here:

$\displaystyle 2 \cdot \vec u \cdot \vec v \leq 2 |\vec u||\vec v|$

Since $\displaystyle \color{blue}|\vec u|,\ \color{red}|\vec v|$ are nonnegative real numbers you can multiply the inequality without changing the relation sign:

$\displaystyle 2 \cdot \vec u \cdot \vec v \cdot \color{blue}|\vec u| \color{black} \cdot \color{red}|\vec v| \color{black} \leq 2 |\vec u||\vec v| \cdot \color{blue}|\vec u| \color{black} \cdot \color{red}|\vec v|$

Now rearrange this inequaltiy:

$\displaystyle 2 \cdot \color{red}|\vec v| \color{black} \cdot \vec u \cdot \color{blue}|\vec u| \color{black} \cdot \vec v \leq 2|\vec u|^2 \cdot |\vec v|^2$

$\displaystyle 2 \cdot (\color{red}|\vec v| \color{black} \cdot \vec u ) \cdot (\color{blue}|\vec u| \color{black} \cdot \vec v) \leq 2|\vec u|^2 \cdot |\vec v|^2$

3. Thank you.