John Stillwell - Naive Lie Theory.pdf - Index of

12 1 The geometry of complex numbers and quaternions
Let v and w be any two complex numbers, and consider their images,
uv and uw under multiplication by u. Thenwehave
distance from uv to uw = |uv − uw|
= |u(v − w)| by the distributive law
= |u||v − w| by multiplicative absolute value
= |v − w| because |u| = 1
= distance from v to w.
In other words, multiplication by u with |u| = 1 is a rigid motion, also
known as an isometry, of the plane. Moreover, this isometry leaves O
fixed, because u × 0 = 0. And if u ≠ 1, no other point v is fixed, because
uv = v implies u = 1. The only motion of the plane with these properties
is rotation about O.
Exactly the same argument applies to quaternion multiplication, at least
as far as preservation of distance is concerned: if we multiply the space
R 4 of quaternions by a quaternion of absolute value 1, then the result is
an isometry of R 4 that leaves the origin fixed. It is in fact reasonable to
interpret this isometry of R 4 as a “rotation,” but first we want to show that
quaternion multiplication also gives a way to study rotations of R 3 . To see
how, we look at a natural three-dimensional subspace of the quaternions.
Pure imaginary quaternions
The pure imaginary quaternions are those of the form
p = bi + cj + dk.
They form a three-dimensional space that we will denote by Ri+Rj+Rk,
or sometimes R 3 for short. The space Ri + Rj + Rk is the orthogonal
complement to the line R1 of quaternions of the form a1, which we will
call real quaternions. From now on we write the real quaternion a1 simply
as a, and denote the line of real quaternions simply by R.
It is clear that the sum of any two members of Ri + Rj + Rk is itself
amemberofRi + Rj + Rk, but this is not generally true of products. In
fact, if u = u 1 i + u 2 j + u 3 k and v = v 1 i + v 2 j + v 3 k then the multiplication
diagram for i, j, andk (Figure 1.2) gives
uv = − (u 1 v 1 + u 2 v 2 + u 3 v 3 )
+(u 2 v 3 − u 3 v 2 )i − (u 1 v 3 − u 3 v 1 )j +(u 1 v 2 − u 2 v 1 )k.