Regarding the first question.
It does involve using vector resolutes.
So what you are being asked to do is find "the vector component of a perpendicular to b". I.e. find the perpendicular resolute. That is the dotted line in the diagram (ignore the |A|cos(theta), not needed).
Lets make where that right angle is point C.
I.e. we are looking for OC.
The vector resolute of
a in the direction of b is
Unlatex'd. OC=[(a.b)/(b.b)]*b
That is representing the vector that is in the same direction as b, but has the lenght from the bottom right most point to where the right angle is, i.e. the projection.
Now we want the vector that is
perpendicular, not
parallel to b.
So that would be the dotted line, now the dotted line will be
Unlatex'd. CA=OA-OC=a-[(a.b)/(b.b)]*b
Hope that helps, probably didn't explain it well.