Orthonormal basis

**pakman** · May 31st 2008, 08:10 PM

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!

**PaulRS** · June 1st 2008, 05:19 AM

Note that: $ 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)

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

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

But we also know that: $ u^t \cdot u_1 = \cos \left( {\measuredangle u^t u_1 } \right) = \tfrac{1} {2} $ and $ 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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