Challenge Level

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 ${\bf b}$.