Problem 1. Show that the cross product defined on R^3 by [x1 x2x3] X [y1 y2 y3]
= [(x2y3 − x3y2), (x3y1 − x1y3), (x1y2 − x2y1)] makes R^3 into analgebra.
We already know that R^3 forms a vector space, so all that needs tobe shown is that
the X operator is bilinear.
Afterwards, show that the cross product is neither commutative norassociative.
A counterexample suffices here. If you want, you can write aprogram that
checks the commutative and associative laws for x, y, z in R^3, andthen simply
generate random integer vectors x, y, z in Z^3 a subset of R^3until those laws fail.
