# Rotate a Vector by Quaternion

I'm trying how to work out how to Rotate a Vertex using Quaternions, using a scientific calculator, or on paper. Exam preparation.

My lecturer has given us this; Quaternion = (-0.5, 0, -0.7071067, 0.5) Vertex = (23, 10, 18)

The way it's been explained to us is like this;

We have a vertex called p

We have a quaternion called q

We store p within a quaternions vector component, we'll call this K

K = (0, p)

Finally we do the normal quaternion multiplication

p' = qKq-1

I'm just trying to work out how I break this down so it's easier to understand so I am able to find out the result. I know how to do quaternion multiplication, but it seems confusing as I only have one w component in the quaternion, and only the x, y,z in the vector.

• To suffices to very the result for infinitesimal rotations. Suppose $q=1+\epsilon$ with $\epsilon$ very small, so $q^{-1}=1-\epsilon+o(\epsilon)$. Then $qKq^{-1}=K+[\epsilon,\,K]-o(\epsilon)$. – J.G. May 2 '19 at 17:50
• en.wikipedia.org/wiki/Quaternion – amd May 2 '19 at 19:00
• I'm not sure I understand what the question is, if you know how to do quaternion multiplication. – Captain Lama May 2 '19 at 19:53

## 1 Answer

We store p within a quaternions vector component, we'll call this $$K = (0, p)$$

[...] but it seems confusing as I only have one w component in the quaternion, and only the x, y,z in the vector.

I have no idea how you are writing and multiplying quaternions, so I'll do it both the ways I think you might be doing it.

If you are doing it with $$i,j,k$$'s then this means you are storing it as $$K=23i+10j+18k$$ and $$q=-0.5 -0.7071067j+0.5k$$

If you are storing it as (scalar, vector) parts and then doing computations that way then it means your quaternion is $$K=(0, (23,10,18))$$ and $$q=(-0.5, (0, -0.7071067, 0.5))$$