1. ## 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!

2. 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