I'm sorry if this has been asked, I've searched around and couldn't find it.
$\displaystyle a + b + c = 0$
where a, b and c are vectors.
Q) Show that a*b = b*c = c*a
Thanks.
If you mean dot product, this isn't true, Take a to be a unit vector, b= -a, c= 0. Then $\displaystyle a\cdot b= |a|^2= 1$ while $\displaystyle a\cdot c= b\cdot c= 0$.
If you mean the cross product, it still isn't true. Take $\displaystyle a= \vec{i}$, $\displaystyle b=-\vec{i}+ \vec{j}$ and $\displaystyle c= -\vec{j}$. Then $\displaystyle a\times b= b\times c= \vec{k}$, but $\displaystyle a\times c= -\vec{k}$.
So what does "*" mean?
It could be the cross product. The last product is c times a, not the other way around.
Edit: And if it is the cross product, you may find it useful to write c = -a-b.
You then need to check that b*c and c*a are both equal to a*b (recall that the cross product is linear in both variables (like any good product), and that x*x=0 for all x).