Exercise [11.02]

Sameed Zahoor · 26 May 2008, 10:07

$q_{1}=t_{1}+u_{1}i+v_{1}j+w_{1}k$
$q_{2}=t_{2}+u_{2}i+v_{2}j+w_{2}k$
Define the operations as follows:
Addition:
$q_{1}+q_{2}=t_{1}+t_{2}+(u_{1}+u_{2})i+(v_{1}+v_{2})j+(w_{1}+w_{2})k$
Multiplication:
$q_{1}q_{2}$
$=t_{1}t_{2}-u_{1}u_{2}-v_{1}v_{2}-w_{1}w_{2}$
$+(t_{1}u_{2}+u_{1}t_{2}+v_{1}w_{2}-w_{1}v_{2})i$
$+(t_{1}v_{2}+v_{1}t_{2}-u_{1}w_{2}+w_{1}u_{2})j$
$+(t_{1}w_{2}+w_{1}t_{2}+u_{1}v_{2}-v_{1}u_{2})k$
(This formula arises when you multiply out $q_{1}q_{2}$ and follow the identities ii=jj=kk=-1, i=jk,j=ki,k=ij ,ij=-ji,jk=-kj and ki=-ik assuming the distributive law)

Exercise [11.02]

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