i have question from vector mathematics,i know that if there is given two vector, for instance $a=\{a_1,a_2,a_3\}$,$b={b_1,b_2,b_3}$; then so called exterior product is determined as $a\wedge b=\frac{1}{2}(ab-ba)$, but I am confused. My question is what is $ab$ or $ba$? is it scalar? in this case $ab=ba$ and $a\wedge b=0$ which does not have any meaning or $ab=(b_1-a_1,b_2-a_2,b_3-a_3)$ or vector? thanks
Update:
consider for example following equation $[a,b][X,Y]=a*x+b*y=c$
then solution is given by following equalities $[ X,Y]=\frac{1}{a\wedge b}[c\wedge b,a\wedge c]$