Transform the vector field:
$$\vec{F}=(\frac{x}{r},\frac{y}{r},\frac{z}{r}), \space\space \space > r=\sqrt{x^2+y^2+z^2}$$
into spherical coordinates.
My attempt: The transformation matrix for transforming a vector(field) from cartesian coordinates to spherical coordinates is given by:
$$\begin{pmatrix}v_r\\v_{\theta} \\ v_{\phi}\end{pmatrix}=\begin{pmatrix}\cos{\phi}\sin{\theta}&\sin{\phi}\sin{\theta}&\cos{\theta}\\\cos{\phi}\cos{\theta}&\sin{\phi}\cos{\theta}&-\sin{\theta}\\-\sin{\phi}&\cos{\phi}&0\end{pmatrix} \begin{pmatrix}v_x\\v_{y} \\ v_{z}\end{pmatrix}\\$$
So my vector field in spherical coordinates would be:
$$\vec{F}_{\text{spherical}}=\begin{pmatrix}\cos{\phi}\sin{\theta}\frac{x}{r}+\sin{\phi}\sin{\theta}\frac{y}{r}+\cos{\theta}\frac{z}{r}\\ \cos{\theta}\cos{\phi}\frac{x}{r}+\sin{\phi}\cos{\theta}\frac{y}{r}-\sin{\theta}\frac{z}{r}\\-\sin{\phi\frac{x}{r}+\cos{\phi}\frac{y}{r}}\end{pmatrix}$$
1st question: Is this correct or do I also have to substitute in $x,y,z$?
2nd question: Why does this method work? How can you arrive at this matrix? Is this like a rotation matrix?