Introduction to Quaternions
Chapter IX.
Formulae and their Application
69. Products of two or more vectors.
1. Two vectors. The relations which exist between the scalars and vectors of the product of two vectors have already been exhibited in Art. 22. We simply extract them:
(a) Sαβ = Sβα.
(b) Vαβ = -Vβα.
(c) αβ + βα = 2Sαβ.
(d) αβ - βα = 2Vαβ.
These we shall quote as formulae (1).
2. We may here add a single conclusion for quaternion products.
Any quaternion, such as αβ may be written as the sum of a scalar and a vector, If therefore q and r be quaternions, we may write
q = Sq + Vq,
r = Sr + Vr;
therefore
qr = SqSr + SqVr + VqSr + VqVr,
and
S·qr = SqSr + S·VqVr,
V·qr = SqVr + SrVq + V·VqVr,
where S·VqVr is the scalar part, and V·VqVr the vector part of the product of the two vectors Vq, Vr.
If now we transpose q and r, and apply (a) and (b) of formulae 1, we get
S·qr = S·rq
(2).
V·qr + V·rq = 2(SqVr + SrVq)
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