Rotation Matrices (Edexcel IGCSE Maths B): Revision Note

Rotation matrices

How do I find rotation matrices?

• Imagine the unit square OABC

  • It has a side-length of 1 unit

  • O is the origin

• The coordinates of A and C as column vectors are

  • $A=\left(\begin{array}{c}1\\ 0\end{array}\right)$ and $C=\left(\begin{array}{c}0\\ 1\end{array}\right)$

• Under a rotation about the origin, A moves to A' and C moves to C' 

  • The matrix, M representing this rotation is $\mathbf{M}=\left(\begin{array}{cc}A\text{'}|& C\text{'}\end{array}\right)$

  • A' and C' are column vectors of the new positions

    • So $\mathbf{M}$ is a 2×2 matrix

  • The points O and B are not needed, as we can draw the rotated square using just A' and C' (as O won't move)

• For example:

  • To find the matrix representing a rotation of 90° anticlockwise about the origin

    • A goes to $A\text{'}=\left(\begin{array}{c}0\\ 1\end{array}\right)$ (on the positive y-axis)

    • C goes to $C\text{'}=\left(\begin{array}{c}-1\\ 0\end{array}\right)$ (on the negative x-axis)

    • $\mathbf{M}=\left(\begin{array}{cc}A\text{'}|& C\text{'}\end{array}\right)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$

  • To find the matrix representing a rotation of 180° about the origin

    • A goes to $A\text{'}=\left(\begin{array}{c}-1\\ 0\end{array}\right)$ (on the negative x-axis)

    • C goes to $C\text{'}=\left(\begin{array}{c}0\\ -1\end{array}\right)$ (on the negative y-axis)

    • $\mathbf{M}=\left(\begin{array}{cc}A\text{'}|& C\text{'}\end{array}\right)=\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)$

      • This is the same as $\mathbf{M}=-\mathbf{I}$ where $\mathbf{I}$ is the identity matrix