Show that for every quaternion z we have

[Lightf]

Homework Statement

Show that for every quaternion z we have:
<br /> $ \overline{z} = \frac{1}{2}(-z-izi - jzj - kzk) $<br /> [\tex]That is the question, I just don&#039;t know how to begin and the &quot; izi - jzj - kzk &quot; confuses me. I need help on how to start this. Thanks a lot :D

 

[Lightf]

I worked it out now.

Let :

z = a + ib + jc + kd

(and z bar = a - ib - jc -kd )

Then just multiply it out.

 

[Mark44]

Homework Statement

Show that for every quaternion z we have:
<br /> \overline{z} = \frac{1}{2}(-z-izi - jzj - kzk) <br />


That is the question, I just don't know how to begin and the " izi - jzj - kzk " confuses me. I need help on how to start this. Thanks a lot :D

Fixed your LaTeX. The closing tag should be [ /tex], not [ \tex] (without the spaces). Also, you don't need the $ characters.

 

[Maybe_Memorie]

Having trouble with the same question.

Any tips?

 

[Lightf]

This is how i did it :

Let :
<br /> z = a + ib + jc + kd<br />
Then sub that into the formula :
<br /> \overline{z} = \frac{1}{2}(-z-izi - jzj - kzk)<br />

And just carefully multiply it out...

<br /> -iz = -ia - i^{2}b - ijc - ikd<br />

<br /> -iz =-ia + b -kc +jd<br />

<br /> -izi = -i^{2}a+ib-kic + jid<br />

<br /> -izi=a+ib-jc-kd<br />

And repeat for -jzj and -kzk and then sub into the formula.

It should all cancel leaving you with : a - ib -jc - kd which is \overline{z}