You can also write (a+b).(a-b) as:
(a+b)·(a-b) = a·a - a·b + b·a - b·b .
Do you know if the dot product is commutative? That will help you to simplify the above expression.
The question is: Suppose the vectors (a+b) and (a-b) are orthogonal. What, if anything, can you conclude about a and b?
So far all I know is that since they are orthogonal, the dot product of (a+b).(a-b)=0. I'm not so sure what to do after that. I've been trying to substitute a=<a1,a2,a3> and b=<b1,b2,b3> to do the dot product and somehow get a.b=0 but no such luck. I don't even think that approach is right. Is there another way to approach this problem?