A useful determinant identity allows the following determinant to be expressed using vector operations, left bracketing bar x_1 | y_1 | z_1 | 1 x_2 | y_2 | z_2 | 1 x_3 | y_3 | z_3 | 1 x_4 | y_4 | z_4 | 1 right bracketing bar = (x_3 - x_1)·[(x_2 - x_1)x(x_4 - x_3)]. Additional interesting determinant identities include left bracketing bar 1 | a | b + c 1 | b | c + a 1 | c | a + b right bracketing bar = 0 (Muir 1960, p.