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

  • Thread starter innuendo999
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In summary, the author is trying to find a quaternion z that can be expressed in the form z = cos(alpha/2) + sin(alpha/2).n, where n is a vector of length 1. The author does not understand the intuition behind this equation and is lost.
  • #1
innuendo999
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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
 
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  • #2


innuendo999 said:
I don't know where to start

Write down a general quaternion z, and write down |z|.
 
  • #3


George Jones said:
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.
 
  • #4


innuendo999 said:
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?
 
  • #5


innuendo999 said:
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 :smile:
 
  • #6


George Jones said:
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
 
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  • #7


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

i've never been in a linear algebra class, I am just working through problems that are in a linear algebra pdf i downloaded :)
 
  • #8


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 [tex]\vec{n}[/tex] 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


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.
 
  • #10


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

1. What is a quaternion?

A quaternion is a mathematical concept that extends the complex numbers and is used in 3D graphics, computer vision, and robotics. It is represented by four numbers, a + bi + cj + dk, where a, b, c, and d are real numbers and i, j, and k are imaginary units.

2. What does it mean for a quaternion to have a magnitude of 1?

A quaternion's magnitude, also known as its norm or length, can be calculated using the Pythagorean theorem. A quaternion with a magnitude of 1 means that it is normalized and can be represented on the unit sphere. This is similar to how complex numbers with a magnitude of 1 lie on the unit circle.

3. How can every quaternion with a magnitude of 1 be expressed?

Every quaternion with a magnitude of 1 can be expressed as a unit quaternion, which has a magnitude of 1 and a real part of 0. These unit quaternions can be written as cos(θ) + sin(θ)(bi + cj + dk), where θ is an angle and b, c, and d are the coordinates on the unit sphere.

4. What is the significance of expressing every quaternion with a magnitude of 1?

Expressing every quaternion with a magnitude of 1 is important because it allows for a simpler representation and easier calculations in certain applications, such as 3D rotations. It also helps to generalize the concept of complex numbers to higher dimensions.

5. Can quaternions with a magnitude other than 1 be expressed in a similar way?

Yes, quaternions with a magnitude other than 1 can be expressed as a combination of a magnitude and a unit quaternion. This is known as the polar form of a quaternion and is similar to how complex numbers can be expressed using polar coordinates.

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