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Thread: Orthonormal basis

  1. #1
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    Orthonormal basis

    Let $\displaystyle \textbf{u}_1$ and $\displaystyle \textbf{u}_2$ form an orthonormal basis for $\displaystyle R^2$ and let $\displaystyle \textbf{u}$ be a unit vector in $\displaystyle R^2$. If $\displaystyle \textbf{u}^T\textbf{u}_1=1/2$, determine the value of $\displaystyle |\textbf{u}^T\textbf{u}_2|$

    Can someone help me begin this problem, thank you!
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  2. #2
    Super Member PaulRS's Avatar
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    Note that: $\displaystyle
    u^t \cdot u_2 = \underbrace {\left\| {u^t } \right\| \cdot \left\| {u_2 } \right\|}_{ = 1} \cdot \cos \left( {\measuredangle u^t u_2 } \right)
    $ (all of our vectors are unit vectors)


    $\displaystyle \measuredangle u^t u_2 $ is one of the angles between the vectors

    $\displaystyle
    u^t \cdot u_2 = \cos \left( {\measuredangle u^t u_2 } \right)
    $

    But we also know that: $\displaystyle
    u^t \cdot u_1 = \cos \left( {\measuredangle u^t u_1 } \right) = \tfrac{1}
    {2}
    $ and $\displaystyle
    u_1 \bot u_2
    $ since it's a orthogonal basis

    Now try to work with a bit of trigonometry and you'll get the answer
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