Why are vectors  and <-b,a> perpendicular? If i have the vectors:
X = {a, b}
Y = {-b, a}
How can I explain that these vectors will always be perpendicular? I know I can prove this very easily via the dot product, but I need to explain it in a layman's way. 

You can argue that $(a,b)\mapsto(-b,a)$ is a $90^°$ rotation be applying this operation twice:
$$(a,b)\quad\mapsto\quad(-b,a)\quad \mapsto\quad(-a,-b).$$
The latter one is obviously the $180^°$ rotated vector to $(a,b)$. As you reached at this point by doing the former operation twice it seems as this would be the half of a $180^°$ turn, and this is a $90^°$ turn. I hope it is layman-terms that such a rotation will make a perpendicular vector.
