# 1.5 Stillwell's Naive Lie Theory rotation by conjugation theorem. Order of multiplication

In section 1.5 of Naive Lie Theory by John Stillwell the rotation by quaternion $$t = \cos\left(\theta\right) + u \cdot \sin\left(\theta\right)$$ is defined as conjugation by $$t$$: $$q \mapsto t^{-1} q t$$ and is said to rotate the set of pure imaginary quaternions $$\mathbb{R}i + \mathbb{R}j + \mathbb{R}k$$ by angle $$2\theta$$.

I understand the proof that this is indeed the rotation, what I'm struggling with is the order of the multiplication which leads to clockwise rotation on the plane orthogonal to $$u$$, which differs from the convention for complex numbers earlier in the book and in general that rotation is clockwise if you look in the direction of the axis around which one rotates. E.x. $$q = 0 + 1i + 0j + 0k$$ would be mapped by $$t = \frac{\sqrt{2}}{2} + 0i + 0j + \frac{\sqrt{2}}{2}k$$ to $$q_{rot} = 0 + 0i - 1j + 0k$$. With a map $$q \mapsto t q t^{-1}$$ the result is as expected: $$q_{rot} = 0 + 0i + 1j + 0k$$

Is it just taking different convention and not stating it or is there something I'm missing?

• What exactly is the problem or what exactly is the question? Oct 24, 2022 at 16:41
• The problem is with the convention, does the author take unpopular one (and in opposite to the rotation realized by complex number multiplication earlier in the book) or is rotation of the whole set assumed to be passive transform (e.x. change of coordinate system)? Oct 24, 2022 at 16:58
• I've never read his book, but normally the derivation of rotation via quaternion would be similar to the complex numbers and work the same, so the variant with the unpopular convention would be likely. Oct 24, 2022 at 17:27

This indeed is the convention. In other words, you get a counterclockwise (aka anticlockwise) if you look from the direction in which the axis of rotation points. This is consistent with the right hand rule, i.e. $$\mathbf{k}\times\mathbf{i}=\mathbf{j}$$ rotates $$\mathbf{i}$$ counterclockwise (looking down from the $$+\mathbf{k}$$-axis) to $$\mathbf{j}$$.
And indeed, if $$t=\exp(\theta\mathbf{u})$$, then $$t\mathbf{x}t^{-1}$$ is the counterclockwise rotation of $$\mathbf{x}$$ around $$\mathbf{u}$$ by $$2\theta$$. That means $$t^{-1}\mathbf{x}t$$ would be the clockwise rotation (looking from the $$+\mathbf{k}$$-axis "downwards"), or a counterclockwise if looking up from below (i.e. in the $$+\mathbf{k}$$-direction).