Question

6. Establish the following quaternion identities:
(a) (x1 + y1i + z1j + w1k)(x2 − y2i − z2j − w2k)
=(x1x2 + y1y2 + z1z2 + w1w2)+(−x1y2 + y1x2 − z1w2 + w1z2)i
+(−x1z2 + z1x2 − w1y2 + y1w2)j +(−x1w2 + w1x2 − y1z2 + z1y2)k
(b) (x2 + y2i + z2j + w2k)(x1 − y1i − z1j − w1k)
=(x1x2 + y1y2 + z1z2 + w1w2)+(x1y2 − y1x2 + z1w2 − w1z2)i
+(x1z2 − z1x2 + w1y2 − y1w2)j +(x1w2 − w1x2 + y1z2 − z1y2)k
(c) The product of a quaternion h = a + bi + cj + dk
and its conjugate h∗ = a − bi − cj − dk is a2 + b2 + c2 + d2.
If q and t are quaternions,then (qt)∗ = t∗q∗.

Answer 1

Using identities:

i.j=k, j.k=i, k.i=j,

j.i=-k, k.j=-i, i.k=-j,

i^2=j^2=k^2=-1