# Show that every quaternion z, where |z|= 1, can be expressed

1. Oct 13, 2009

### innuendo999

hi, and thanks for reading. hh, and this isn't homework, its just something i've been wondering about.

i've been flicking through a linear algebra book, i'm trying to learn it by myself, and i've come across this question which has completely stumped me:

show that every quaternion z, where |z|= 1, can be expressed in the form z = cos(alpha/2) + sin(alpha/2).n, where n is a vector of length 1

I don't know where to start, but more importantly, i don't understand the intuition behind it. Anybody care to explain? thanks

Last edited: Oct 13, 2009
2. Oct 13, 2009

### George Jones

Staff Emeritus
Re: Quaternions

Write down a general quaternion z, and write down |z|.

3. Oct 13, 2009

### innuendo999

Re: Quaternions

Thanks for the reply. I've that much done. And I know that a^2 + b^2 + c^2 + d^2 = 1. But, that's where I'm lost.

4. Oct 13, 2009

### George Jones

Staff Emeritus
Re: Quaternions

I assume that b, c, and d are the coefficients of i, j, and k respectively.

What does

a^2 + b^2 + c^2 + d^2 = 1

and

b^2 + c^2 + d^2 >= 0

5. Oct 13, 2009

### Zorba

Re: Quaternions

Innuendo, are you Irish?
We got this exact same question for homework in Linear Algebra, to hand in today... /suspicious

6. Oct 13, 2009

### innuendo999

Re: Quaternions

yes, b, c and d are the coefficients of i, j and k

it says that a is less than or equal to 1?

so, n = i + j + k, then b, c and d = sin(alpha/2)? and a = cos(alpha/2)? i can see that much, but i can't see how to get one from the other

Last edited: Oct 13, 2009
7. Oct 13, 2009

### innuendo999

Re: Quaternions

i've never been in a linear algebra class, im just working through problems that are in a linear algebra pdf i downloaded :)

8. Oct 13, 2009

### Zorba

Re: Quaternions

Ahh well if that's the case, then I can give you a few hints since I solved it myself.

Think about how to construct $$\vec{n}$$ in such a way that satisfies the question.
Think about when Sin/Cos is defined.
Think about that equation George Jones gave you and there's a certain trig identity that may allow you to manipulate it.

9. Oct 13, 2009

### iasc

Re: Quaternions

Hey Zorba I'm from Ireland and had to hand up this question in class today.
You doin maths in trinity?

As for the the question I couldn't quite get it.
Sorry.

10. Oct 22, 2009

### Zorba

Re: Quaternions

Aye, I'm in Trinity, but doing TP though.