Matrices that turn and enlarge the plane
Matrices that turn and enlarge the plane
IP 3
8 marks
The matrix \(\mathbf{R}=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\) maps the point \((x,y)\) to the point with position \(\mathbf{R}\begin{pmatrix} x \\ y \end{pmatrix}\).
- Show that \(\mathbf{R}\) maps \(P(x,y)\) to \(Q(-y,x)\). By comparing \(OP\) with \(OQ\) and the gradients of \(OP\) and \(OQ\), explain why the map is a rotation of \(90^\circ\) about \(O\) (anticlockwise).[3]
- Find \(\mathbf{R}^2\) and \(\mathbf{R}^4\), and describe the maps they represent.[2]
- Find the matrix \(\mathbf{M}\) that maps \((1,0)\) to \((3,4)\) and \((0,1)\) to \((-4,3)\). Show that \(\mathbf{M}\) multiplies every length from \(O\) by \(5\).[3]
Solution:
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Answer:(b) \(\mathbf{R}^2=-\mathbf{I}\) (half turn), \(\mathbf{R}^4=\mathbf{I}\) (c) \(\mathbf{M}=\begin{pmatrix} 3 & -4 \\ 4 & 3 \end{pmatrix}\)
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