Team Project Useful Formulas for Two and more Vectors, prove (12) ?? (16), which are often useful in practical work, and illustrate each formula with tow examples. 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). Formula (15) is called Lagrange??sidentity.

(12) (13) (14) (a x b) x (cx d) a x b = (a.a)(b.b) - (a.b) bx (ex d) = (b.d)c - (b.c)d (15) (a x b) (16) = (a b d)c (a b c)d (cx d) = (a c)(b d) (a.d)(b.c) - (a b c) = (b ca) = (ca b) = -(c b a) = (a c b)