Rotation Matrices

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Rotation matrices are matrices which are used to describe the rotation of a rigid body from one orientation to another.

In R3 they form a 3x3 matrix.

$$ \mathbf{R} = \begin{bmatrix} \bhat{u_x} & \bhat{v_x} & \bhat{w_x} \\ \bhat{u_y} & \bhat{v_y} & \bhat{w_y} \\ \bhat{u_z} & \bhat{v_z} & \bhat{w_z} \end{bmatrix} $$

such that:

$$ \b{Ra} = \b{b} $$

\(\b{a}\) and \(\b{b}\) are 1x3 vectors

Combining Rotations

Two successive rotations represented by \(\b{R_1}\) and \(\b{R_2}\) can be represented by a single rotation matrix \(\b{R_3}\) where:

$$ \b{R_3} = \b{R_2} \b{R_1} $$

Pay careful attention to the order of the matrix multiplication, successive rotation matrices are multiplied on the left.

How To Find The Rotation Matrix Between Two Coordinate Systems

Suppose I have the frame with the following unit vectors defining the first coordinate system \( X1Y1Z1 \):

$$ X1=\begin{bmatrix}x_x\\x_y\\x_z\end{bmatrix} \quad Y1=\begin{bmatrix}y_x\\y_y\\y_z\end{bmatrix} \quad Z1=\begin{bmatrix}z_x\\z_y\\z_z\end{bmatrix} $$

And a second coordinate system \( X2Y2Z2 \) defined by the unit vectors:

$$ X2=\begin{bmatrix}x_x\\x_y\\x_z\end{bmatrix} \quad Y2=\begin{bmatrix}y_x\\y_y\\y_z\end{bmatrix} \quad Z2=\begin{bmatrix}z_x\\z_y\\z_z\end{bmatrix} $$

The rotation matrix \( R \) which rotates objects from the first coordinate system \(X1Y1Z1\) into the second coordinate system \(X2Y2Z2\) is:

$$ R = \begin{bmatrix} X1 \cdot X2 & X1 \cdot Y2 & X1 \cdot Z2\\ Y1 \cdot X2 & Y1 \cdot Y2 & Y1 \cdot Z2\\ Z1 \cdot X2 & Z1 \cdot Y2 & Z1 \cdot Z2\\ \end{bmatrix} $$

\( \cdot \) is the matrix dot product
and everything else as above

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