Suppose the vector product

a \times b \neq 0

. Define a sequence of vectors

b_{0}, b_{1}, b_{2} \dots

b_{0} = b

and

b_{n + 1} = a \times b_{n}

Show that $b_{n} \to 0$ as $n \to \infty$ if the length $‖ a ‖$ is less than one.

If $‖ a ‖ = 1$ and $‖ b_{1} ‖ = r$ find the directions of the first six vectors in the sequence in relation to the vector $a$ and draw a diagram showing these vectors. What happens to the sequence? Describe the surface on which the sequence of vectors from $b_{1}$ onwards lies.

Note: You need to know that the vector product $a \times b$ 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 $a$ and $b$ .