Convertion from and to spherical - cartesian

[M. next]

I googled it, and it says:
\dot{x}=\dot{r}sinθcos∅ + (rcosθcos∅)\dot{θ} - (rsinθsin∅)\dot{∅}
.
.
and so on for \dot{y} & \dot{z}

And then they wrote "We will also need the inverse transformation obtained by solving the equations above w.r.t \dot{r}, \dot{θ}, and \dot{∅}

for example they got:
\dot{r}=sinθcos∅\dot{x}+sinθsin∅\dot{y} + cosθ\dot{z}
.
.
and so on, my question how did they deduced the latter from the former equations?

Here's the link if you want to see them clearer:
http://www.physics.sc.edu/~yar/Phys701_2009/homework/hw9_solutions.pdf

Just what's the procedure?
Thanks

 

[tiny-tim]

Hi M. next!

… for example they got:
\dot{r}=sinθcos∅\dot{x}+sinθsin∅\dot{y} + cosθ\dot{z}
…
Just what's the procedure?

It's just algebra (plus standard trigonometric identities) …

for example, the r' component of the RHSs is r'(sin²θcos²∅ + sin²θsin²∅ + cos²θ) = r'

 

[M. next]

Hmm.. I mean how do we get r' from (x', y', z')?

 

[M. next]

I solved three equations with three unknowns and it worked but after so much effort! Thanks anyways Tim.