Composition of Rotation matrix - how?

Question

I am, at the moment, learning about rotation matrices, and it seems confusing to me how this is possible:

$$R_A^C = R_A^B \cdot R_B^C$$

So.. $R_A^C$ must for a $2\times2$ matrix be defined as $[x^a \cdot x^b ~~~~ x^a \cdot x^b~~ ;~~ y^a \cdot y^b ~~~~ y^a \cdot y^b ]$

I don't see how using this standard the multiplication stated above will give the same matrix as for $R_A^C$. Some form for clarification would be helpful here.

Answer 1

Let me rephrase to a notation I am more used to

Rotation by angle $\phi$: $$R_\phi = \left[\begin{array}{cc}a_\phi & b_\phi\\ -b_\phi & a_\phi\end{array}\right]$$

Rotation by angle $\theta$:

$$R_\theta = \left[\begin{array}{cc}a_\theta & b_\theta\\ -b_\theta & a_\theta \end{array}\right]$$

Now the composition (using matrix multiplication) becomes:

$$R_{\phi+\theta} = R_\phi R_\theta = \left[\begin{array}{cc}a_\theta a_\phi - b_\theta b_\phi & a_\theta b_\phi + b_\theta a_\phi \\ -b_\theta a_\phi -a_\theta b_\phi & -b_\theta b_\phi + a_\theta a_\phi \end{array}\right]$$

Then you can look at the 4 elements and see that upper left = lower left, so we have a new $a$ variable, and the upper right is equal to the lower right but with switched sign, so that is our new $b$.

Answer 2

If by $R_A^B$ you mean a rotation of an angle $A\hat OB$, then if we call that angle $\phi$ and we call $\theta$ the angle $B\hat OC$, then the matrix representing those rotations are:

\begin{align*} R_A^B={}&\left[\begin{array}{cc} \cos\phi & -\sin\phi \\ \sin\phi & \cos\phi \end{array}\right], \\ R_B^C={}&\left[\begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array}\right]. \end{align*}

Let us now compute the product of these, and use some trig identities:

\begin{align*} R_B^C\cdot R_A^B={}&\left[\begin{array}{cc} \cos\phi & -\sin\phi \\ \sin\phi & \cos\phi \end{array}\right]\cdot\left[\begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array}\right]={} \\ {}={}&\left[\begin{array}{cc} \cos\phi\cos\theta-\sin\phi\sin\theta & -\cos\phi\sin\theta-\sin\theta\cos\phi \\ \sin\phi\cos\theta+\cos\phi\sin\theta & -\sin\phi\sin\theta+\cos\phi\cos\theta \end{array}\right]={} \\ {}={}&\left[\begin{array}{cc} \cos(\phi+\theta) & -\sin(\phi+\theta) \\ \sin(\phi+\theta) & \cos(\phi+\theta) \end{array}\right]. \end{align*}

And that is exactly the rotation by $\theta+\phi$, i.e. $R_A^C$.