isometries-rotation-compose

Exercise.routineBeyond 21-242 Let

𝑅_{𝑥}

be the quarter turn of

ℝ^{3}

about the

𝑥_{1}

-axis with

𝑅_{𝑥} 𝐞_{2} = 𝐞_{3}

, and

𝑅_{𝑦}

the quarter turn about the

𝑥_{2}

-axis with

𝑅_{𝑦} 𝐞_{3} = 𝐞_{1}

. Find the axis and angle of

𝑅_{𝑦} 𝑅_{𝑥}

Solution.

Uses. Theorem 8.4.27 (Euler’s rotation theorem), Definition 8.4.26 (Rotation)

Hint 1. Write down the matrices:

𝑅_{𝑥}

fixes

𝐞_{1}

and sends

𝐞_{2} \mapsto 𝐞_{3} \mapsto - 𝐞_{2}

Hint 2. The product is a rotation by Remark 8.4.18. Find a kernel vector of

𝑃 - 𝐼_{3}

and use the trace.

Sketch.

𝑅_{𝑥} = (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & 0 \end{matrix})

𝑅_{𝑦} = (\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ - 1 & 0 & 0 \end{matrix})

𝑃 =

𝑅_{𝑦} 𝑅_{𝑥} =

(\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & - 1 \\ - 1 & 0 & 0 \end{matrix})

𝑃 (1, 1, - 1) =

(1, 1, - 1)

and

tr 𝑃 = 0

, so

𝑃

turns through

2 𝜋 / 3

about

(1, 1, - 1) / \sqrt{3}

$𝑅_{𝑥}$ fixes $𝐞_{1}$ and sends $𝐞_{2} \mapsto 𝐞_{3}$ , $𝐞_{3} \mapsto - 𝐞_{2}$ ; $𝑅_{𝑦}$ fixes $𝐞_{2}$ and sends $𝐞_{3} \mapsto 𝐞_{1}$ , $𝐞_{1} \mapsto - 𝐞_{3}$ . So

𝑅_{𝑥} = (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & 0 \end{matrix}), 𝑅_{𝑦} = (\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ - 1 & 0 & 0 \end{matrix}), 𝑃 = 𝑅_{𝑦} 𝑅_{𝑥} = (\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & - 1 \\ - 1 & 0 & 0 \end{matrix}) .

$𝑃$ is a product of rotations, hence a rotation (its columns $- 𝐞_{3}, 𝐞_{1}, - 𝐞_{2}$ are orthonormal and $det 𝑃 =$ $det 𝑅_{𝑦} det 𝑅_{𝑥} =$ $1$ ). Solving $𝑃 𝐯 = 𝐯$ : $𝑣_{2} = 𝑣_{1}$ , $- 𝑣_{3} = 𝑣_{2}$ , $- 𝑣_{1} = 𝑣_{3}$ , so the axis is spanned by $(1, 1, - 1)$ . Since $tr 𝑃 = 0 = 1 + 2 cos 𝜃$ , $cos 𝜃 = - 1 / 2$ , and $𝑃$ turns through $2 𝜋 / 3$ about $𝐚 = \frac{1}{\sqrt{3}} (1, 1, - 1)$ . Two quarter turns about perpendicular axes compose to a third of a turn about a diagonal.