Prove (13)–(16), which are often useful in practical work, and illustrate each formula with two examples. For (13) choose Cartesian coordinates such that d = [d₁, 0, 0] and c = [c₁, c₂, 0]. Show that each side of (13) then equals [-b₂c₂d₁, b₁c₂d₁, 0], and give reasons why the two sides are then equal in any Cartesian coordinate system. For (14) and (15) use (13).

(13) b × (c × d) = (b • d)c - (b • c)d

(14) (a × b) × (c × d) = (a b d)c - (a b c)d

(15) (a × b) • (c × d) = (a • c)(b • d) - (a • d)(b • c)

(16) (a b c) = (b c a) = (c a b) = -(c b a) = -(a c b)