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

[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

 

[George Jones]

I don't know where to start

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

 

[innuendo999]

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

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.

 

[George Jones]

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.

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

say about a^2, and thus about a?

 

[Zorba]

Hi, and thanks for reading. Oh, and this isn't homework, it's just something I've been wondering about.

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

 

[innuendo999]

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

say about a^2, and thus about a?

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

 

[innuendo999]

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

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

 

[Zorba]

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.

 

[iasc]

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

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

 

[Zorba]

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