Suppose the vector product a ×b ¹ 0. Define a sequence of vectors b0, b1, b2¼ by b0=b and bn+1=a×bn
Show that bn ® 0 as n® ¥ if
the length |a| is less than one.
If |a|=1 and |b1|=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 b1 onwards lies.
Note: You need to know that the vector product a ×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.