Quaternion Rotations: Show R2∘R1 Is a Rotation

In summary, the conversation discusses the process of writing a quaternion z of length 1 in the form z = cos(\alpha/2) + sin(\alpha/2)n, for some number α and some vector n, |n| = 1. It also explores the composition of two rotations in 3D space and shows that this composition is also a rotation around some vector through some angle. This is demonstrated using the quaternionic terms and the length of w = w_2w_1 is shown to be 1. The previous problem is then used to solve the current problem.
  • #1
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Homework Statement



3.
[STRIKE](a) Show that every quaternion [tex]z[/tex] of length 1 can be written in the
form [tex]z = cos(\alpha/2) + sin(\alpha/2)n[/tex], for some number α and some vector [tex]n[/tex], [tex]|n| = 1.[/tex][/STRIKE]

(b) Consider two rotations of the 3d space: the rotation [tex]R_1[/tex] through [tex]\alpha_1[/tex] around the vector [tex]n_1[/tex] and the rotation [tex]R_2[/tex] through [tex]\alpha_2[/tex] around the vector [tex]n_2[/tex]. Define a new transformation, the composition [tex]R_2 \circ R_1[/tex], in the usual way: it takes a vector v, rotates it using [tex]R_1[/tex], and then rotates the result using [tex]R_2: R_2 \circ R_1(v) = R_2(R_1(v))[/tex]. Show that this composition is also a rotation around some vector through some angle. (Hint: in quaternionic terms, [tex]R1[/tex] brings v to [tex]w_1v\overline{w_1}[/tex]), and [tex]R_2[/tex] brings the result to

[tex]z = cos(\alpha/2) + sin(\alpha/2)n[/tex]

[tex]w_2w_1v\overline{w_1}\overline{w_2} = (w_2w_1)v(\overline{w_2w_1})[/tex]

Now, check that [tex]w = w_2w_1[/tex] is of length 1, and use the previous problem.)

Homework Equations



[tex]w_2w_1v\overline{w_1}\overline{w_1}\overline{w_2} = (w_2w_1)v(\overline{w_2w_1})[/tex].

The Attempt at a Solution



I don't understand the question, if someone could explain what I must do that would be really helpful!
 
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  • #2
Vladimir actually did this out for us in class sometime last week.

You could probably approach this using the method of the optional question from last weeks homework.

(off topic, I know we're in the same math class, but who are you??)
 
  • #3
I'll check my notes again and try again ;)

(I'm Fergus - You doing pure maths?)
 
  • #4
I stopped writing my notes just as he started explaining the rotations.. Typical.

I'm stumped with this tbh.

(I'm Paul - I am, and I know you are too because I saw your thread about the computation quiz ;) )
 

Related to Quaternion Rotations: Show R2∘R1 Is a Rotation

1. What are quaternion rotations?

Quaternion rotations are a mathematical concept used to describe rotations in three-dimensional space. They are represented by four numbers, called quaternions, and can be used to rotate objects or points in space.

2. How do quaternion rotations work?

Quaternion rotations use a combination of complex numbers and vectors to represent rotations in three-dimensional space. These rotations can be applied to an object or point by multiplying the quaternion by the object's or point's coordinates.

3. What is the significance of R2∘R1 in quaternion rotations?

R2∘R1 represents the composition of two quaternion rotations, with R1 being applied first and then R2 being applied to the result. This is important because quaternion rotations are not commutative, meaning the order in which they are applied matters.

4. How can it be shown that R2∘R1 is a rotation?

In order to show that R2∘R1 is a rotation, we can use a geometric proof. We can show that the composition of two rotations results in a new rotation by examining the angle and axis of rotation of R2∘R1. If the angle and axis remain the same, then it is a rotation.

5. Why are quaternion rotations useful?

Quaternion rotations are useful because they are a more efficient and accurate way to represent rotations in three-dimensional space compared to other methods, such as Euler angles. They are also used in computer graphics and robotics to rotate objects and points in a more intuitive and precise manner.

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