A vector in cartesian space can be represented in two ways. In component form, we write the vector in terms of its components:

$r=r_x$hat$(i)+r_y$hat$(y)+r_z$hat$(z)=($:$r_x,r_y,r_z$:$)$

Or we can express a vector in terms of its magnitude and direction, where the magnitude is

$|r|=r=\sqrt[\phantom{2}]{r_x^2+r_y^2+r_z^2}$

and the azimuthal angle $($angle in the $x-y$ plane$)$ is

$=ta{n}^{-1}(\frac{r_y}{r_x})$

and the polar angle $($angle from the positive $z-$axis$)$ is

$=co{s}^{-1}(\frac{r_z}{|r|})$

We can invert these relations to get $r_x,r_y$ and $r_z$ in terms of the magnitude and angles:

$r_x=rsincos,r_y=rsinsin,r_z=rcos$

We'd like to be able to translate between these two forms of vectors.

Tasks

In the next cell, create and document two functions called comp$\_$to$\_$mag$\_$direc, and mag$\_$direc$\_$to$\_$comp

Each function should take in up to three arguments, and return up to three values

comp$\_$to$\_$mag$\_$direc should take in an x component, a y component, and optionally a z component. It should return the magnitude, azimuthal angle, and potentially the polar angle $($if the user provided a value for $z),$ in that order. The angles should be given in radians.

mag$\_$direc$\_$to$\_$comp will do the exact opposite. It should take in a magnitue, an azimuthal angle $($in radians$),$ and optionally a polar angle $($also in radians$).$ It should return at least the $x$ and $y$ components, and optionally the $z$ component if the polar angle was given.

Notes and Hints

Note that $$ can be any angle from $0$ to $2,$ but arctan will only give angles from $-\frac{}{2}$ to $+\frac{}{2}.$ You could account for this with an if block to conditionally add $$ to the angle. But even better is the math module's atan$2,$ which can take care of everything for you. Give it a google to see how it works.

You will need to set default values for arguments. Be sure to document these arguments appropriately according to the Numpydoc standard.

You may call the arguments to your functions whatever you want. I$\text{'}$d suggest x $,$ y$,$ and z for comp$\_$to$\_$mag$\_$direc and mag, phi, and theta for mag$\_$direc$\_$to$\_$comp, but the tests are indifferent to what you call them.

Big hint: None makes for a good default value for the optional third parameters; users will never use it intentionally, so you won't confuse it for a valid input.