John Stillwell - Naive Lie Theory.pdf - Index of

1.4 Consequences of multiplicative absolute value 13
This relates the quaternion product uv to two other products on R 3 that are
well known in linear algebra: the inner (or “scalar” or “dot”) product,
u · v = u 1 v 1 + u 2 v 2 + u 3 v 3 ,
and the vector (or “cross”) product
∣ i j k ∣∣∣∣∣
u × v =
u 1 u 2 u 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.
∣v 1 v 2 v 3
In terms of the scalar and vector products, the quaternion product is
uv = −u · v + u × v.
Since u · v is a real number, this formula shows that uv is in Ri + Rj + Rk
only if u · v = 0, that is, only if u is orthogonal to v.
The formula uv = −u · v + u × v also shows that uv is real if and only
if u × v = 0, that is, if u and v have the same (or opposite) direction. In
particular, if u ∈ Ri + Rj + Rk and |u| = 1 then
u 2 = −u · u = −|u| 2 = −1.
Thus every unit vector in Ri + Rj + Rk is a “square root of −1.” (This, by
the way, is another sign that H does not satisfy all the usual laws of algebra.
If it did, the equation u 2 = −1 would have at most two solutions.)
Exercises
The cross product is an operation on Ri+Rj+Rk because u×v is in Ri+Rj+Rk
for any u,v ∈ Ri + Rj + Rk. However, it is neither a commutative nor associative
operation, as Exercises 1.4.1 and 1.4.3 show.
1.4.1 Prove the antisymmetric property u × v = −v × u.
1.4.2 Prove that u × (v × w)=v(u · w) − w(u · v) for pure imaginary u,v,w.
1.4.3 Deduce from Exercise 1.4.2 that × is not associative.
1.4.4 Also deduce the Jacobi identity for the cross product:
u × (v × w)+w × (u × v)+v × (w × u)=0.
The antisymmetric and Jacobi properties show that the cross product is not completely
lawless. These properties define what we later call a Lie algebra.