"""simple docstring"""


UpperCAmelCase__  		=						tuple[float, float, float]
UpperCAmelCase__  		=						tuple[float, float, float]






def     __UpperCAmelCase					(			lowercase   ,lowercase		):







  """simple docstring"""
  _UpperCAmelCase						 =      end_pointa[0] - end_pointa[0]
  _UpperCAmelCase						 =      end_pointa[1] - end_pointa[1]
  _UpperCAmelCase						 =      end_pointa[2] - end_pointa[2]
  return (x, y, z)






def     __UpperCAmelCase					(			lowercase   ,lowercase		):







  """simple docstring"""
  _UpperCAmelCase						 =      ab[1] * ac[2] - ab[2] * ac[1]  # *i
  _UpperCAmelCase						 =      (ab[0] * ac[2] - ab[2] * ac[0]) * -1  # *j
  _UpperCAmelCase						 =      ab[0] * ac[1] - ab[1] * ac[0]  # *k
  return (x, y, z)






def     __UpperCAmelCase					(			lowercase   ,lowercase		):







  """simple docstring"""
  return tuple(round(lowercase   ,lowercase		) for x in vector		) == (0, 0, 0)






def     __UpperCAmelCase					(			lowercase   ,lowercase   ,lowercase   ,lowercase = 10		):







  """simple docstring"""
  _UpperCAmelCase						 =      create_vector(lowercase   ,lowercase		)
  _UpperCAmelCase						 =      create_vector(lowercase   ,lowercase		)
  return is_zero_vector(get_ad_vectors_cross(lowercase   ,lowercase		)   ,lowercase		)



"""simple docstring"""


from __future__ import annotations

from random import choice






def     __UpperCAmelCase					(			lowercase		):







  """simple docstring"""
  return choice(lowercase		)






def     __UpperCAmelCase					(			lowercase   ,lowercase		):







  """simple docstring"""
  _UpperCAmelCase						 =      random_pivot(lowercase		)

  # partition based on pivot
  # linear time
  _UpperCAmelCase						 =      [e for e in lst if e < pivot]
  _UpperCAmelCase						 =      [e for e in lst if e > pivot]

  # if we get lucky, pivot might be the element we want.
  # we can easily see this:
  # small (elements smaller than k)
  # + pivot (kth element)
  # + big (elements larger than k)
  if len(lowercase		) == k - 1:
    return pivot
  # pivot is in elements bigger than k
  elif len(lowercase		) < k - 1:
    return kth_number(lowercase   ,k - len(lowercase		) - 1		)
  # pivot is in elements smaller than k
  else:
    return kth_number(lowercase   ,lowercase		)


if __name__ == "__main__":
    import doctest

    doctest.testmod()
