Force vector in a new coordinate frame?

[theBEAST]

Homework Statement

http://img811.imageshack.us/img811/9092/captureykj.png

The Attempt at a Solution

The answer is also in the image above. I have no clue how to start this question. Could anyone be so kind to give me a hint on how I should approach this question? Thanks!

 

[gabbagabbahey]

Homework Statement

http://img811.imageshack.us/img811/9092/captureykj.png

The Attempt at a Solution

The answer is also in the image above. I have no clue how to start this question. Could anyone be so kind to give me a hint on how I should approach this question? Thanks!

The statement "$\mathbf{f}$ is measured as $\begin{bmatrix} 1 \\ -1\end{bmatrix}\text{N}$ in the $u$, $v$ coordinate basis" means that $\mathbf{f} = (1 \text{N})\mathbf{e}_u +(-1\text{N}) \mathbf{e}_v$. So, if you can do a little trig to express $\mathbf{e}_u$ & $\mathbf{e}_v$ in terms of $\mathbf{e}_x$ & $\mathbf{e}_y$, you can express $\mathbf{f}$ in the $x$, $y$ coordinate basis.

Hint: $\mathbf{A} \cdot \mathbf{B} = ||\mathbf{A}||||\mathbf{B}|| \cos \theta$, where $\theta$ is the angle between the two vectors.