John Stillwell - Naive Lie Theory.pdf - Index of

14 1 The geometry of complex numbers and quaternions
1.5 Quaternion representation of space rotations
A quaternion t of absolute value 1, like a complex number of absolute value
1, has a “real part” cos θ and an “imaginary part” of absolute value sinθ,
orthogonal to the real part and hence in Ri + Rj + Rk. This means that
t = cosθ + usinθ,
where u is a unit vector in Ri+Rj+Rk, and hence u 2 = −1 by the remark
at the end of the previous section.
Such a unit quaternion t induces a rotation of Ri + Rj + Rk, though
not simply by multiplication, since the product of t and a member q of
Ri + Rj + Rk may not belong to Ri + Rj + Rk. Instead, we send each
q ∈ Ri+Rj+Rk to t −1 qt, which turns out to be a member of Ri+Rj+Rk.
To see why, first note that
t −1 = t/|t| 2 = cosθ − usinθ,
by the formulas for q −1 and q in Section 1.3.
Since t −1 exists, multiplication of H on either side by t or t −1 is an
invertible map and hence a bijection of H onto itself. It follows that the
map q ↦→ t −1 qt, called conjugation by t, is a bijection of H. Conjugation by
t also maps the real line R onto itself, because t −1 rt = r for a real number
r; hence it also maps the orthogonal complement Ri + Rj + Rk onto itself.
This is because conjugation by t is an isometry, since multiplication on
either side by a unit quaternion is an isometry.
It looks as though we are onto something with conjugation by t =
cosθ + usinθ, and indeed we have the following theorem.
Rotation by conjugation. If t = cosθ + usinθ, whereu∈ Ri + Rj + Rk
is a unit vector, then conjugation by t rotates Ri + Rj + Rk through angle
2θ about axis u.
Proof. First, observe that the line Ru of real multiples of u is fixed by the
conjugation map, because
t −1 ut =(cos θ − usinθ)u(cos θ + usinθ)
=(ucos θ − u 2 sinθ)(cos θ + usinθ)
=(ucos θ + sinθ)(cos θ + usinθ) because u 2 = −1
= u(cos 2 θ + sin 2 θ)+sinθ cosθ + u 2 sinθ cosθ
= u also because u 2 = −1.