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Math Help - Simple expression manipulation.

  1. #1
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    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, 2 (\vec{u} \cdot \vec{v}) \lVert \vec{u} \rVert \lVert \vec{v} \rVert. Is that not correct? Or are they the same expression?
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  2. #2
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    Quote Originally Posted by scorpion007 View Post
    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
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  3. #3
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    Thank you.
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