Suppose the vector product ${\bf a} \times {\bf b}\neq {\bf 0}$.
Define a sequence of vectors ${\bf b_0},\ {\bf b_1},\ {\bf
b_2}\ldots $ by ${\bf b_0}={\bf b}$ and ${\bf b_{n+1}}={\bf
a}\times {\bf b_n}$
Show that ${\bf b_n} \rightarrow 0$ as $n\rightarrow \infty$ if the
length $|{\bf a}|$ is less than one.
If $|{\bf a}|=1$ and $|{\bf b_1}|=r$ find the
directions of the first six vectors in the sequence in relation to
the vector ${\bf a}$ and draw a diagram showing these vectors. What
happens to the sequence? Describe the surface on which the sequence
of vectors from ${\bf b_1}$ onwards lies.
Note: You need to know that the vector product ${\bf a} \times {\bf
b}$ is the product of the magnitudes of the vectors times the sine
of the angle between the vectors and it is a vector perpendicular
to ${\bf a}$ and ${\bf b}$.