First note that your angle is being measured in a clockwise sense with $\theta=0$ being the $y$-axis.

the thing you want to calculate is the magnitude of the component of velocity along a particular axis right? Then the way I would do it is not look at $u$ and $v$ but instead look at Speed "$s$" and Direction "$\theta$". If you want to know the magnitude of the component of velocity along a direction $\phi$, compute $s \cos(\theta - \phi)$.

You could do it with the $u$'s and $v$'s. Then let $c = \cos(\phi)$ and redefine $s$ to be $\sin(\phi)$. Then the answer would be $us+vc$. The idea being that you are rotating your vector by $\phi$ so that what was in the $\phi$ direction is now pointing along the $y$ axis. Then you take the $y$ component to get what was the component along the $\phi$ direction before the rotation.

Note that what you did wrong is you have $\pi - \phi$ instead of $\phi$.

First note that your angle is being measured in a clockwise sense with $\theta=0$ being the $y$-axis.

the thing you want to calculate is the magnitude of the component of velocity along a particular axis right? Then the way I would do it is not look at $u$ and $v$ but instead look at the Speed "$s$" and Direction "$\theta$" that you are also given. If you want to know the magnitude of the component of velocity along a direction $\phi$, compute $s \cos(\theta - \phi)$.

You could do it with the $u$'s and $v$'s. Then let $c = \cos(\phi)$ and redefine $s$ to be $\sin(\phi)$. Then the answer would be $us+vc$. The idea being that you are rotating your vector by $\phi$ so that what was in the $\phi$ direction is now pointing along the $y$ axis. Then you take the $y$ component to get what was the component along the $\phi$ direction before the rotation.

Note that what you did wrong is you have $\pi - \phi$ instead of $\phi$.