Rotation Matrices (Edexcel IGCSE Maths)

Spec point — 5G

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

  • $A = ( \begin{gathered} 1 \\ 0 \end{gathered} )$ and $C = ( \begin{gathered} 0 \\ 1 \end{gathered} )$

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

  • The matrix, M representing this rotation is $M = ( \begin{gathered} A ' \vert & C ' \end{gathered} )$

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

    • So $M$ 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 $A ' = ( \begin{gathered} 0 \\ 1 \end{gathered} )$ (on the positive y-axis)
    • C goes to $C ' = ( \begin{gathered} - 1 \\ 0 \end{gathered} )$ (on the negative x-axis)
    • $M = ( \begin{gathered} A ' \vert & C ' \end{gathered} ) = ( \begin{gathered} 0 - 1 \\ 1 0 \end{gathered} )$

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

    • A goes to $A ' = ( \begin{gathered} - 1 \\ 0 \end{gathered} )$ (on the negative x-axis)

    • C goes to $C ' = ( \begin{gathered} 0 \\ - 1 \end{gathered} )$ (on the negative y-axis)

    • $M = ( \begin{gathered} A ' \vert & C ' \end{gathered} ) = ( \begin{gathered} - 1 & 0 \\ 0 - 1 \end{gathered} )$

      • This is the same as $M=-I$ where* *$I$ is the identity matrix

Worked Example The matrix $M$ represents a rotation of 270° anticlockwise about the origin.

Work out $M$.

Answer:

A rotation of 270° anticlockwise is the same as a rotation of 90° clockwise

Consider how the points A and C on the unit square are transformed

The point A $( \begin{gathered} 1 \\ 0 \end{gathered} )$ moves to A' $( \begin{gathered} 0 \\ - 1 \end{gathered} )$

The point C $( \begin{gathered} 0 \\ 1 \end{gathered} )$ moves to C' $( \begin{gathered} 1 \\ 0 \end{gathered} )$

The transformation matrix is given by $M = ( \begin{gathered} A ' \vert & C ' \end{gathered} )$

$\mathrm{M} = ( \begin{gathered} 0 1 \\ - 1 & 0 \end{gathered} )$