# Equality of products.

• May 6th 2011, 03:06 AM
Equality of products.
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.
• May 6th 2011, 05:59 AM
Prove It
Are you using an asterix to represent dot product, cross product, componentwise multiplication, or something else entirely?
• May 6th 2011, 06:10 AM
HallsofIvy
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?
• May 6th 2011, 07:28 AM
HappyJoe
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).
• May 7th 2011, 08:46 PM
i was referring to the cross product, sorry if I wasn't clear enough :L
• May 7th 2011, 11:35 PM
HappyJoe
Have you solved the problem yet?
• May 7th 2011, 11:53 PM
Prove It
How many dimensions are the vectors?
• May 8th 2011, 02:01 AM