Cross Product‎ ‎ ‎

Question

Given u times v = <2,1,4> find (u-2v)times(u+4v)

Answer 1

u x v = <2, 1, 4>

(u - 2v) x (u + 4v) = (u x u) + 4(u x v) - 2(v x u) - 8(v x v) =

< 0, 0, 0 > + 4 < 2, 1, 4 > - 2 <-2, -1, -4 > - 8 < 0, 0, 0 > = < 12, 6, 24 >

Note that u x u is the zero vector and u x v = - (v x u)

Answer 2

One key to remember here is that cross products are finding a vector that is perpendicular to both of the given vectors; this new vector has the length that is equal to the area that would be produced by the parallelogram produced by the original vectors. If you cross multiply a vector by itself, there's no single vector perpendicular to the original vector and the parallelogram would have no area. This gives a zero vector.

Answer 3

u•v = <2, 1, 4>; (u - 2v)•(u + 4v) = u•u + 4u•v - 2v•u - 8v•v = 0 + 6u•v - 0 = <12, 6, 24>