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

In the first part of the question where $|{\bf a}|< 1$ 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 ${\bf a}={\bf i}$ and
${\bf b_1} =r{\bf j}$