Module 1 : A Crash Course in Vectors
Lecture 2 : Coordinate Systems

Note that (see figure) the angle that $\hat\rho$makes an angle $\theta$with the x-axis ( $\hat\imath$) and $\pi/2-\theta$with the y-axis ( $\hat\jmath$). Similarly, the unit vector $\hat\theta$makes $\pi/2+\theta$with the x-axis and $\theta$with the y-axis. Thus

\begin{eqnarray*} A_\rho = \vec A\cdot\hat\rho &=& A_x\hat\imath\cdot\hat\rho + ... ...(\pi/2+\theta)+A_y\cos\theta\\ &=& -A_x\sin\theta+A_y\cos\theta \end{eqnarray*}

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