Show that for every quaternion z we have

[Lightf]

Homework Statement

Show that for every quaternion z we have:
$$\overline{z} = \frac{1}{2}(-z-izi - jzj - kzk) <br /> [\tex]That is the question, I just don't know how to begin and the " izi - jzj - kzk " confuses me. I need help on how to start this. Thanks a lot :D$$

 

[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:
$$\overline{z} = \frac{1}{2}(-z-izi - jzj - kzk)$$


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

And just carefully multiply it out...

$$-iz = -ia - i^{2}b - ijc - ikd$$

$$-iz =-ia + b -kc +jd$$

$$-izi = -i^{2}a+ib-kic + jid$$

$$-izi=a+ib-jc-kd$$

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 $$\overline{z}$$