Simple expression manipulation.

**scorpion007** · March 8th 2009, 04:58 AM

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, $2 (\vec{u} \cdot \vec{v}) \lVert \vec{u} \rVert \lVert \vec{v} \rVert$ . Is that not correct? Or are they the same expression?

**earboth** · March 8th 2009, 11:32 AM

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, $2 (\vec{u} \cdot \vec{v}) \lVert \vec{u} \rVert \lVert \vec{v} \rVert$ . Is that not correct? Or are they the same expression?

I'll start here:

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

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

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

Now rearrange this inequaltiy:

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

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

**scorpion007** · March 8th 2009, 03:34 PM

Thank you.

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