Question

Let (R 3 , ×) be the set of 3d vectors equipped with the operation of vector crossproduct. Which of the following properties does this operation satisfy (give proofs in all cases)?

(a) has identity element(s)? (If so, determine all identity elements.)

(b) has idempotent element(s)? (If so, determine all idempotent elements.)

(c) commutative?

(d) associative?

Answer 1

(a). let M = (a,b,c) and N = (p,q,r) be 2 arbitrary vectors in R³ . Then, M x N = (br-cq,cp-ar,aq-bp). Now, if M x N = M, then we must have br-cq = a ,cp-ar = b and aq-bp = c, which need not be true when a,b,c are arbitrary real numbers. For example, if M =(1,0,0) and if N (p,q,r), then M x N=( 0,-r,q) so that M x N cannot be equal to M for any values of p,q, r. Hence, the set (R³ , ×) does not have any identity element.

(b). The cross product of any vector in R³ with itself is the zero vector as (a,b,c) x (a,b,c) = (0,0,0). Hence the set (R³ , ×) does not have any idempotent element other than the zero vector (0,0,0).

( c). Let M = (1,2,3) and N = (1,3,2). Then M x N = (-5,1,1) and N X M = (5,-1,-1) so that M X N is not equal to N x M. Thus, cross product is not commutative in R³.

(d). Let M=(1,2,3),N=(1,3,2) and P=(2,3,1).Then, M x (N x P)=(1,2,3) x (-3,3,-3)=(-15,-6,-9) and (M x N) x P = (-5,1,1) x( 2,3,1) = (-2,7,-17). Thus, cross product is not associative in R³.