1. Rotation matrices

Definition 1.1 . A rotation matrix is an orthogonal matrix of determinant $+1$, rather than $-1$. (Note: The identity matrix is considered a rotation matrix.)

Proposition 1.2 . (a) The product of two $n \times n$ rotation matrices is a rotation matrix. (b) The inverse of a rotation matrix is a rotation matrix. (In fact, it's the transpose.)

1.3 Rotation matrices in R$^2$

Our notation will be $R_{\theta} = \left[\begin{array}{rr} \cos \theta& \sin \theta\\ -\sin \theta&\cos \theta \end{array}\right]$, giving a rotation about the origin through an angle $\theta$, measured counterclockwise.

Proposition 1.4 . All $2 \times 2$ rotation matrices are of this form.

1.5 Rotation matrices in R$^3$

By $R_{\theta}^{x\rightarrow y}$ we will mean the rotation matrix that gives a rotation about the $z$-axis by an angle $\theta$ measured starting from the $x$-axis and rotating towards the $y$-axis. Its effect on the $x,y$-plane is the same as the effect of $R_{\theta}$. Because $(1,0,0)$ and $(0,1,0)$ are moved in the $x,y$-plane as if $R_{\theta}$ were being applied, and because $(0,0,1)$ is left fixed, $R_{\theta}^{x\rightarrow y}$ $=$ $\left[\begin{array}{rrr}\cos \theta&\sin \theta& 0\\ - \sin \theta& \cos \theta&0\\ 0&0&1\end{array}\right]$.

$R_{\theta}^{y\rightarrow x}$ is the inverse of $R_{\theta}^{x\rightarrow y}$. Thus $R_{\theta}^{y\rightarrow x}$ gives the rotation about the $z$-axis by an angle $\theta$ measured starting from the $y$-axis and rotating towards the $x$-axis.

$R_{\theta}^{x\rightarrow z}$, $R_{\theta}^{z\rightarrow x}$, $R _ {\theta}^{y\rightarrow z}$, and $R_{\theta}^{z\rightarrow y}$ are defined similarly.

1.6 Remarks.

(1)

Not all $3 \times 3$ rotation matrices are of these forms, but all can be obtained as products of matrices of these forms.

(2)

The word axis can mean two things: (i) a coordinate axis, i.e., the $x$-axis, $y$-axis, or $z$-axis; (ii) the axis of a rotation in R$^3$, i.e., a line of points that are left fixed (are not moved) by the rotation. It is a fact that for every $3 \times 3$ matrix the corresponding rotation in R$^3$ has an axis. (See Problem F-.)

(3)

Because a rotation matrix has positive determinant, the rotation it describes preserves not only angles (like all orthogonal transformations) but also orientation, and so it preserves cross products. Because the rows of a rotation matrix are the images of i$,$j$,$k, they have the same cross-product relations as i$,$j$,$k. In particular, since i$\times$   j$=$   k, the third row of a rotation matrix is the cross product of the first and second rows. (What about columns?)