How Do Quaternions Rotate Vectors in 3D Space?

[IgF]

Homework Statement

Hi, I have just started learning quarternions and I understand all the simple stuff like adding/subtracting, multiplication, inverse, magnitude and finding the complex conjugate but now I am trying to get my head around rotation and I can't seem to get anywere. So below is the question and then i'll show what I have found (im certain it is very wrong ).
I haven't yet tried to find the induced matrix as I am really unsure of this so any knowledge on that would also be very much appreciated. Sorry for the length but thanks in advance to anyone that can offer assistance.Create the view quaternion, V. The x, y, z, components are taken from the
“lookat” vector.

The scalar (real) component of the quaternion V is 0, hence this is a pure
quaternion.

Let “lookat” = < 0, 0, 1 >

Rotate the quaternion V by the following axis and angle.
Axis = < 0, 1, 0 >
Angle = pi / 4

You will need to encode the axis and angle in a quaternion, R. Find R.

Rotate the quaternion V by R, and hence find the resulting rotation W.

Where W = RVR*

Create the new “lookat” vector by getting the x, y, z, components form the

resulting W quaternion.

What is the resulting “lookat” vector?

Find the induced matrix for R.

The Attempt at a Solution

quarternion V = 0 + 0i + 0j + 1k

R =
(cos(45 / 2) + sin(45 / 2)0) = 0
(cos(45 / 2) + sin(45 / 2)1) = 1.3066
(cos(45 / 2) + sin(45 / 2)0) = 0

R = 0 + 0i + 1.3066j + 0k

W = RVR*

W = 0 + 0i + 0j - 1.705k

'look at' vector = < 0, 0, -1.71>

 

[nate808]

Hello, thank you for reaching out for assistance with your quaternion rotation problem. It seems like you have a good understanding of the basic operations of quaternions, but may need some clarification on how to apply them to rotation.

Firstly, let's review the properties of quaternions. They are a four-dimensional extension of complex numbers, with the form a + bi + cj + dk, where a, b, c, and d are real numbers and i, j, and k are imaginary units. The key property of quaternions is that they can be used to represent rotations in three-dimensional space. This is because they can be written in the form q = cos(theta/2) + sin(theta/2)u, where theta is the angle of rotation and u is a unit vector representing the axis of rotation.

Now, let's address the specific problem at hand. We are given a "lookat" vector, which represents the direction in which an observer is looking. We want to rotate this vector by an angle of pi/4 (45 degrees) around an axis of <0,1,0>. To do this, we need to create a rotation quaternion R that encodes this information. We can do this by following these steps:

1. Normalize the axis vector. This is important because we want our quaternion to represent a unit rotation, and the axis vector may not be a unit vector. In this case, our axis vector is already normalized, so we can skip this step.

2. Calculate the angle of rotation, theta. In this case, theta = pi/4.

3. Use the formula q = cos(theta/2) + sin(theta/2)u to create the quaternion R. Since theta = pi/4, we have R = cos(pi/8) + sin(pi/8)u.

4. Substitute in the values for cosine and sine, and the axis vector <0,1,0>, to get R = 0.9239 + 0.3827i + 0j + 0.3827k.

Now that we have our rotation quaternion R, we can apply it to the "lookat" vector using the formula W = RVR*. This will give us the new "lookat" vector after rotation. We can also find the induced matrix for R by using the formula M = 2q*q^-1 - I, where q is the quaternion and q^-1 is