


"""simple docstring"""
def 				_snake_case      (   __snake_case							:      str						, __snake_case							:      str   ):




   """simple docstring"""







   _lowerCamelCase	:					str   =		len(__snake_case   )
   _lowerCamelCase	:					Union[str, Any]   =		len(__snake_case   )
   _lowerCamelCase	:					int   =		[[False for _ in range(m + 1   )] for _ in range(n + 1   )]
   _lowerCamelCase	:					Union[str, Any]   =		True
   for i in range(__snake_case   ):
      for j in range(m + 1   ):
         if dp[i][j]:
            if j < m and a[i].upper() == b[j]:
               _lowerCamelCase	:					Tuple   =		True
            if a[i].islower():
               _lowerCamelCase	:					Tuple   =		True
   return dp[n][m]


if __name__ == "__main__":
 import doctest

 doctest.testmod()





"""simple docstring"""
from __future__ import annotations

from numpy import array, cos, cross, floataa, radians, sin
from numpy.typing import NDArray

def 				_snake_case      (   __snake_case							:      float						, __snake_case							:      float						, __snake_case							:      bool = False   ):




   """simple docstring"""







   if radian_mode:
      return [magnitude * cos(__snake_case   ), magnitude * sin(__snake_case   )]
   return [magnitude * cos(radians(__snake_case   )   ), magnitude * sin(radians(__snake_case   )   )]

def 				_snake_case      (   __snake_case							:      NDArray[floataa]						, __snake_case							:      NDArray[floataa]						, __snake_case							:      float = 10**-1   ):




   """simple docstring"""







   _lowerCamelCase	:					NDArray[floataa]   =		cross(__snake_case						, __snake_case   )
   _lowerCamelCase	:					float   =		sum(__snake_case   )
   return abs(__snake_case   ) < eps


if __name__ == "__main__":
 # Test to check if it works
 UpperCAmelCase				=						array(
     [
         polar_force(718.4, 180 - 30),
         polar_force(879.54, 45),
         polar_force(100, -90),
     ]
 )

 UpperCAmelCase				=						array([[0, 0], [0, 0], [0, 0]])

 assert in_static_equilibrium(forces, location)

 # Problem 1 in image_data/2D_problems.jpg
 UpperCAmelCase				=						array(
     [
         polar_force(30 * 9.81, 15),
         polar_force(215, 180 - 45),
         polar_force(264, 90 - 30),
     ]
 )

 UpperCAmelCase				=						array([[0, 0], [0, 0], [0, 0]])

 assert in_static_equilibrium(forces, location)

 # Problem in image_data/2D_problems_1.jpg
 UpperCAmelCase				=						array([[0, -2000], [0, -1200], [0, 1_5600], [0, -1_2400]])

 UpperCAmelCase				=						array([[0, 0], [6, 0], [10, 0], [12, 0]])

 assert in_static_equilibrium(forces, location)

 import doctest

 doctest.testmod()

