A rotation followed by a rotation is a rotation

In preparation for answering exercise 2.6.3 in Gilbert Strang’s Linear Algebra and Its Applications, Third Edition, I wanted to derive in detail the effect of a reflection followed by a reflection, a reflection followed by a rotation, and a rotation followed by a reflection. For completeness this post recapitulates in a little more detail the argument in section 2.6 that a rotation followed by a rotation is a rotation.

Assume that we have a matrix that rotates vectors through the angle \theta and a second matrix that rotates vectors through the angle \varphi. Intuitively we’d conclude that the effect of applying both matrices in succession is to rotate vectors through the angle \theta+\varphi. Let’s prove this conjecture.

The first rotation is represented by the matrix

Q_\theta = \begin{bmatrix} \cos \theta&-\sin \theta \\ \sin \theta&\cos \theta \end{bmatrix}

and the second rotation is represented by the matrix

Q_\varphi = \begin{bmatrix} \cos \varphi&-\sin \varphi \\ \sin \varphi&\cos \varphi \end{bmatrix}

The effect of the two rotations is thus represented by the product of the matrices:

\begin{bmatrix} \cos \varphi&-\sin \varphi \\ \sin \varphi&\cos \varphi \end{bmatrix} \begin{bmatrix} \cos \theta&-\sin \theta \\ \sin \theta&\cos \theta \end{bmatrix}

= \begin{bmatrix} \cos \varphi \cos \theta - \sin \varphi \sin \theta&-\cos \varphi \sin \theta - \sin \varphi \cos \theta \\ \sin \varphi \cos \theta + \cos \varphi \sin \theta&-\sin \varphi \sin \theta + \cos \varphi \cos \theta \end{bmatrix}

= \begin{bmatrix} \cos \varphi \cos \theta - \sin \varphi \sin \theta&-(\sin \varphi \cos \theta + \cos \varphi \sin \theta) \\ \sin \varphi \cos \theta + \cos \varphi \sin \theta&\cos \varphi \cos \theta - \sin \varphi \sin \theta \end{bmatrix}

Given the trigonometric identities

\begin{array}{rcl} \cos (\varphi + \theta)&=&\cos \varphi \cos \theta - \sin \varphi \sin \theta \\ \sin (\varphi + \theta)&=&\sin \varphi \cos \theta + \cos \varphi \sin \theta \end{array}

we have

\begin{bmatrix} \cos \varphi \cos \theta - \sin \varphi \sin \theta&-(\sin \varphi \cos \theta + \cos \varphi \sin \theta) \\ \sin \varphi \cos \theta + \cos \varphi \sin \theta&\cos \varphi \cos \theta - \sin \varphi \sin \theta \end{bmatrix}

= \begin{bmatrix} \cos (\varphi + \theta)&-\sin (\varphi + \theta) \\ \sin (\varphi + \theta)&\cos (\varphi + \theta) \end{bmatrix}

But this matrix represents Q_{\varphi+\theta}, the rotation of all vectors in the x-y plane through the angle \varphi+\theta. We therefore have Q_\varphi Q_\theta = Q_{\varphi+\theta}.

Note that if we first rotate through the angle \varphi and then through the angle \theta this is equivalent to rotating through the angle \theta and then through the angle \varphi. We therefore have Q_\theta Q_\varphi = Q_\varphi Q_\theta.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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