# Simple expression manipulation.

• Mar 8th 2009, 04:58 AM
scorpion007
Simple expression manipulation.
I'm only concerned about the expressions highlighted in red.

http://img293.imageshack.us/img293/9408/math2.png

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?
• Mar 8th 2009, 11:32 AM
earboth
Quote:

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

http://img293.imageshack.us/img293/9408/math2.png

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:

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

$\displaystyle 2 \cdot (\color{red}|\vec v| \color{black} \cdot \vec u ) \cdot (\color{blue}|\vec u| \color{black} \cdot \vec v) \leq 2|\vec u|^2 \cdot |\vec v|^2$
• Mar 8th 2009, 03:34 PM
scorpion007
Thank you.