Angle of rotation by pure unit quaternions(fundamental unit quaternions)

Question

From the versor definition here, i understand that versor is a unit quaternion and also a pure quaternion with its scalar part zero. So, versor is a pure unit quaternion.

I think that fundamental quaternion units $i,j,k$ are also versors(!).

From the angle-axis notation $\mathbf {q} =\mathbf cos(\alpha/2) + sin(\alpha/2) \vec{v} $.

To get a pure quaternion, we need a $0$ for cosinus part. For $cos(\alpha/2)=0$ we could use $\alpha=180$ so $sin(90)=1$. So, if we have $q=i$ then we could say its a 180 degrees rotation.

Are there anything wrong with this?

Answer 1

Proof: $i,j,k$ are 180 degrees rotation around the axis defined by them.

1. Fundamental quaternion units are versors by definition. Versors are pure unit quaternions.

2. The angle-axis notation of a quaternion $\mathbf {q} =\mathbf cos(\alpha/2) + sin(\alpha/2) \vec{v} $.

3. To get a pure quaternion, we need a $0$ for cosinus part. For $cos(\alpha/2)=0$ we have $\alpha/2=90,270,...$ so $\alpha=180$ so $sin(90)=1$ we could take $\alpha=180$. So, if we have $q=i$ then we could say $i,j,k$ are 180 degrees($\pi$) rotation. $\blacksquare $