



'''simple docstring'''





def        a_     (							_lowerCAmelCase       ,_lowerCAmelCase       )      ->      str:
  if a < 0 or b < 0:
    raise ValueError('the value of both inputs must be positive'       )
  __lowerCamelCase       :       Dict           = str(bin(_lowerCAmelCase       )       )[2:]  # remove the leading "0b"
  __lowerCamelCase       :       Tuple           = str(bin(_lowerCAmelCase       )       )[2:]
  __lowerCamelCase       :       Any           = max(len(_lowerCAmelCase       )       ,len(_lowerCAmelCase       )       )
  return "0b" + "".join(
      str(int('1' in (char_a, char_b)       )       )
      for char_a, char_b in zip(a_binary.zfill(_lowerCAmelCase       )       ,b_binary.zfill(_lowerCAmelCase       )       )       )


if __name__ == "__main__":
   import doctest

   doctest.testmod()











'''simple docstring'''





_UpperCamelCase											=						tuple[float, float, float]
_UpperCamelCase											=						tuple[float, float, float]

def        a_     (							_lowerCAmelCase       ,_lowerCAmelCase       )      ->      Vectorad:
  __lowerCamelCase       :       Any           = end_pointa[0] - end_pointa[0]
  __lowerCamelCase       :       str           = end_pointa[1] - end_pointa[1]
  __lowerCamelCase       :       Tuple           = end_pointa[2] - end_pointa[2]
  return (x, y, z)

def        a_     (							_lowerCAmelCase       ,_lowerCAmelCase       )      ->      Vectorad:
  __lowerCamelCase       :       List[str]           = ab[1] * ac[2] - ab[2] * ac[1]  # *i
  __lowerCamelCase       :       Dict           = (ab[0] * ac[2] - ab[2] * ac[0]) * -1  # *j
  __lowerCamelCase       :       List[Any]           = ab[0] * ac[1] - ab[1] * ac[0]  # *k
  return (x, y, z)

def        a_     (							_lowerCAmelCase       ,_lowerCAmelCase       )      ->      bool:
  return tuple(round(_lowerCAmelCase       ,_lowerCAmelCase       ) for x in vector       ) == (0, 0, 0)


def        a_     (							_lowerCAmelCase       ,_lowerCAmelCase       ,_lowerCAmelCase       ,_lowerCAmelCase = 10       )      ->      bool:
  __lowerCamelCase       :       str           = create_vector(_lowerCAmelCase       ,_lowerCAmelCase       )
  __lowerCamelCase       :       Dict           = create_vector(_lowerCAmelCase       ,_lowerCAmelCase       )
  return is_zero_vector(get_ad_vectors_cross(_lowerCAmelCase       ,_lowerCAmelCase       )       ,_lowerCAmelCase       )






