import matplotlib.pyplot as plt

import math

import astropy.coordinates

#Takes a $3$D vector, and returns a tuple of the x$,$ y$,$ and z components

def spherical$\_$to$\_$components$($magnitude$,$ bearing, trajectory$)$:

return astropy.coordinates.spherical$\_$to$\_$cartesian$($magnitude$,$ math.radians$($trajectory$),$ math.radians$($bearing$))$

#Takes the x$,$ y$,$ and z components of a $3$D vector, and returns a tuple of magnitude, bearing, and trajectory

def components$\_$to$\_$spherical$($x$,$ y$,$ z$)$:

magnitude, trajectory, bearing $=$ astropy.coordinates.cartesian$\_$to$\_$spherical$($x$,$ y$,$ z$)$

return magnitude, math.degrees$($bearing$.$to$\_$value$()),$ math.degrees$($trajectory$.$to$\_$value$())$

#Takes two $3$D vectors $($each specified by magnitude, bearing, and trajectory$)$ and returns a

#tuple representing the sum of the two vectors

def add$\_$spherical$\_$vectors$($magnitude$1,$ bearing$1,$ trajectory$1,$ magnitude$2,$ bearing$2,$ trajectory$2)$:

x$1,$ y$1,$ z$1=$ spherical$\_$to$\_$components$($magnitude$1,$ bearing$1,$ trajectory$1)$

x$2,$ y$2,$ z$2=$ spherical$\_$to$\_$components$($magnitude$2,$ bearing$2,$ trajectory$2)$

return components$\_$to$\_$spherical$($x$1+$ x$2,$ y$1+$ y$2,$ z$1+$ z$=============================================$start with the code provided above and please use this approach below$========$ import matplotlib.pyplot as plt

class SimpleDTSim:

def $\_\_$init$\_\_($self$)$:

# You might want to initialize the simulation here, but that means that you will initialize when the

# object is created, rather than on every simulation run

self.x $=0$

self.a $=0$

self.result $=[]$

def initialize$($self$,$ starting$\_$x$,$ a$)$:

self.x $=$ starting$\_$x

self.a $=$ a

self.result $=[$self$.$x$]$

def observe$($self$)$:

self.result.append$($self$.$x$)$

def update$($self$)$:

self.x $=$ self.a $*$ self.x

def runsim$($self$,$ x$,$ a$,$ steps$)$:

self.initialize$($x$,$ a$)$

for t in range$($steps$)$:

self.update$()$

self.observe$()$

plt$.$plot$($self$.$result$)$

plt$.$show$()$

if $\_\_$name$\_\_==\text{'}\_\_$main$\_\_\text{'}$:

sim $=$ SimpleDTSim$()$

sim.runsim$(1,1\mathrm{.}1,30)$