11.3 The Cross Product of Two Vectors

Cross product A vector in space that is orthogonal to two given vectors If u=u 1 i+u 2 j+u 3 k and v=v 1 i+v 2 j+v 3 k then u x v = (u 2 v 3 -u 3 v 2 )i – (u 1 v 3 -u 3 v 1 )j + (u 1 v 2 -u 2 v 1 )

Find the cross product u=6i + 2j +k and v=i + 3j -2k

Properties of cross products 1) u x v = -(v x u) 2) u x (v+w)=(u x v)+(u x w) 3) c(u x v) = (cu) x v = u x (cv) 4) u x 0 = 0 x u = 0 5) u x u =0 6) u ∙ (v x w) = (u x v)∙ w

Geometric Properties u x v is orthogonal to both u and v u x v =0 if and only if u and v are scalar multiples Area of the parallelogram having u and v as adjacent sides =

Orthogonal Unit Vectors A unit vector orthogonal to both u and v