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Math Help - Orthonormal basis

  1. #1
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    Orthonormal basis

    Let \textbf{u}_1 and \textbf{u}_2 form an orthonormal basis for R^2 and let \textbf{u} be a unit vector in R^2. If \textbf{u}^T\textbf{u}_1=1/2, determine the value of |\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: <br />
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)<br />
(all of our vectors are unit vectors)


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

    <br />
u^t  \cdot u_2  = \cos \left( {\measuredangle u^t u_2 } \right)<br />

    But we also know that: <br />
u^t  \cdot u_1  = \cos \left( {\measuredangle u^t u_1 } \right) = \tfrac{1}<br />
{2}<br />
and <br />
u_1  \bot u_2 <br />
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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