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Exercise [11.03]

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Shaun Culver

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Joined: 25 Feb 2008, 13:32
Posts: 105
Location: Cape Town, South Africa

Exercise [11.03]

If
$q^{-1}=\bar{q}\left(q\bar{q}\right)^{-1}$

Where
q=t+u i+v j+w k

$\bar{q}=t-u i-v j-w k$

Then

$q^{-1}=\bar{q}\left(q\bar{q}\right)^{-1}$

$\frac{1}{(t+u i+v j+w k)}=\frac{(t-u i-v j-w k)}{(t+u i+v j+w k)(t-u i-v j-w k)}$

$\frac{1}{(t+u i+v j+w k)(t-u i-v j-w k)}=\frac{1}{(t+u i+v j+w k)(t-u i-v j-w k)}$
$\frac{1}{t^2+u^2 +v^2 +w^2}=\frac{1}{t^2+u^2 +v^2 +w^2}$

1=1

The inverse quaternion is equal to the quaternion conjugate multiplied by the inverse of the norm.

_________________
“Our imagination is stretched to the utmost, not, as in fiction, to imagine things which are not really there, but just to comprehend those things which are there.”

Last edited by Shaun Culver on 28 May 2008, 17:57, edited 3 times in total.

Edited

25 May 2008, 01:33

Sameed Zahoor

Joined: 12 Mar 2008, 10:57
Posts: 69
Location: India

Re: Exercise [11.3]

The conjugate is defined by,
$\overline{q}=t-ui-vj-wk$
The inverse satisfies
$q^{-1}q=qq^{-1}=1$
We define $q^{-1}=\frac{\overline{q}}{(q\overline{q})}$
This definition works because
$qq^{-1}=q\frac{\overline{q}}{(q\overline{q})}=\frac{q\overline{q}}{(q\overline{q})}=\frac{t^2+u^2+v^2+w^2}{t^2+u^2+v^2+w^2}=1$ and
$q^{-1}q=\frac{\overline{q}}{(q\overline{q})}q=\frac{\overline{q}q}{(q\overline{q})}=\frac{t^2+u^2+v^2+w^2}{t^2+u^2+v^2+w^2}=1$
(because $q\overline{q}=\overline{q}q$ ;see[11.2])
P.S.Shaun,your argument should go like this.
$q^{-1}=\frac{1}{(t+ui+vj+wk)}=\frac{(t-ui-vj-wk)}{(t+ui+vj+wk)(t-ui-vj-wk)}$
$=\frac{\overline{q}}{t^2+u^2+v^2+w^2}=\frac{\overline{q}}{(q\overline{q})}$

26 May 2008, 10:32

Shaun Culver

Site Admin

Joined: 25 Feb 2008, 13:32
Posts: 105
Location: Cape Town, South Africa

Re: Exercise [11.3]

I've modified the solution slightly.

Is there something wrong with my argument?

_________________
“Our imagination is stretched to the utmost, not, as in fiction, to imagine things which are not really there, but just to comprehend those things which are there.”

26 May 2008, 17:33

Sameed Zahoor

Joined: 12 Mar 2008, 10:57
Posts: 69
Location: India

Re: Exercise [11.3]

Initially the argument did have an error,but you seem to have spotted it and corrected it.

27 May 2008, 09:25

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Exercise [11.03]

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