import math

ZERO = (0, 0)


def dot(x1, y1, x2, y2):
    """Returns the dot product of two vectors."""
    return (x1 * x2) + (y1 * y2)


def length(x, y):
    """Returns the magnitude of a vector."""
    return math.sqrt((x * x) + (y * y))

def normalize(x, y):
    """Returns the unit vector pointed in the same direction as [x, y]."""
    _len = length(x, y)
    return x / _len, y / _len


def lnormal(x, y):
    """Returns the left-hand normal of a vector."""
    return -y, x


def rnormal(x, y):
    """Returns the right-hand normal of a vector."""
    return y, -x


def reverse(x, y):
    """Returns a vector pointing in the opposite direction."""
    return -x, -y


def scale(x, y, factor):
    """Scale a vector by a given factor."""
    return x * factor, y * factor


def add(x1, y1, x2, y2):
    """Add two vectors."""
    return x1 + x2, y1 + y2


def sub(x1, y1, x2, y2):
    """Subtract a vector from another."""
    return x1 - x2, y1 - y2


def vmin(x1, y1, x2, y2):
    """Return the minimum [x, y] components of the given vectors."""
    return min(x1, x2), min(y1, y2)


def vmax(x1, y1, x2, y2):
    """Return the maximum [x, y] components of the given vectors."""
    return max(x1, x2), max(y1, y2)


def is_equal(x1, y1, x2, y2):
    """Returns true if two unit vectors are pointed in either the same or
    opposite directions.
    """
    return math.isclose(abs(dot(x1, y1, x2, y2)), 1, rel_tol=1e-05)