This is a bit like a geometric sequence except that the elements are vectors, the 'common ratio' is vector $a$ and the multiplication is vector products.

In the first part of the question where $|a|<1$ use the fact that $|\sin \theta |<1$ to show that the magnitudes of the vectors in the sequence decrease.

In the second half of the question take $a=i$ and $b_1=rj$