4.03e Successive transformations: matrix products

5

1. Write down the \(2 \times 2\) matrices which represent the following plane transformations:
   1. an anticlockwise rotation about the origin through an angle \(\alpha\);
   2. a reflection in the line \(y = x \tan \left( \frac { 1 } { 2 } \beta \right)\).
   3. A reflection in the \(x - y\) plane in the line \(y = x \tan \left( \frac { 1 } { 2 } \theta \right)\) is followed by a reflection in the line \(y = x \tan \left( \frac { 1 } { 2 } \phi \right)\). Show that the composition of these two reflections (in this order) is a rotation and describe this rotation fully.

9 The plane transformation \(T\) is the composition (in this order) of

• a reflection in the line \(y = x \tan \frac { 1 } { 8 } \pi\); followed by
• a shear parallel to the \(y\)-axis, mapping \(( 1,0 )\) to \(( 1,2 )\); followed by
• a clockwise rotation through \(\frac { 1 } { 4 } \pi\) radians about the origin; followed by
• a shear parallel to the \(x\)-axis, mapping \(( 0,1 )\) to \(( - 2,1 )\).

Determine the matrix \(\mathbf { M }\) which represents \(T\), and hence give a full geometrical description of \(T\) as a single plane transformation.

11

1. 1. Write down the matrix which represents a rotation through an angle \(\alpha\) anticlockwise about the origin.
   2. Show that the plane transformation given by the matrix $$\left( \begin{array} { c c } \cos \theta + \sin \theta & - ( \sin \theta - \cos \theta ) \\ \sin \theta - \cos \theta & \cos \theta + \sin \theta \end{array} \right)$$ is the composition of a rotation, \(R\), and a second transformation, \(S\). Describe both \(R\) and \(S\) fully.
2. 1. Write down the matrix which represents a reflection in the line \(y = x \tan \frac { 1 } { 2 } \beta\). For \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\), the plane transformation \(T\) is given by the matrix $$\left( \begin{array} { c c } 1 + \cos 2 \theta & \sin 2 \theta \\ \sin 2 \theta & - 1 - \cos 2 \theta \end{array} \right)$$
   2. Show that \(T\) is the composition of a reflection and an enlargement, and describe these transformations in full.
   3. Find also the values of \(\theta\) for which \(T\) is an area-preserving transformation.

The transformation \(S\) is represented by the matrix \(\mathbf{M} = \begin{bmatrix} 1 & -6 \\ 2 & 7 \end{bmatrix}\) The transformation \(T\) is a reflection in the line \(y = x\sqrt{3}\) and is represented by the matrix \(\mathbf{N}\) The point \(P(x, y)\) is transformed first by \(S\), then by \(T\) The result of these transformations is the point \(Q(3, 8)\) Find the coordinates of \(P\) Give your answers to three decimal places. [5 marks]

A transformation of the \(x\)-\(y\) plane is represented by the matrix \(\begin{pmatrix} \cos \theta & 2 \sin \theta \\ 2 \sin \theta & -\cos \theta \end{pmatrix}\), where \(\theta\) is a positive acute angle.

1. Write down the image of the point \((2, 3)\) under this transformation. [2]
2. You are given that this image is the point \((a, 0)\). Find the value of \(a\). [5]

The transformation \(T\) in the plane consists of a reflection in the line \(y = x\), followed by a translation in which the point \((x, y)\) is transformed to the point \((x + 1, y - 2)\), followed by an anticlockwise rotation through \(90°\) about the origin.

1. Find the \(3 \times 3\) matrix representing \(T\). [6]
2. Show that \(T\) has no fixed points. [3]

The \(2 \times 2\) matrix A represents a rotation by \(90°\) anticlockwise about the origin. The \(2 \times 2\) matrix B represents a reflection in the line \(y = -x\). The matrix B is given by $$\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$$

1. Write down the matrix representing A. [1]
2. The \(2 \times 2\) matrix C represents a rotation by \(90°\) anticlockwise about the origin, followed by a reflection in the line \(y = -x\). Compute the matrix C and describe geometrically the single transformation represented by C. [3]

The transformation \(P\) is an enlargement, centre the origin, with scale factor \(k\), where \(k > 0\) The transformation \(Q\) is a rotation through angle \(\theta\) degrees anticlockwise about the origin. The transformation \(P\) followed by the transformation \(Q\) is represented by the matrix $$\mathbf{M} = \begin{pmatrix} -4 & -4\sqrt{3} \\ 4\sqrt{3} & -4 \end{pmatrix}$$

1. Determine
   1. the value of \(k\),
   2. the smallest value of \(\theta\) [4]

A square \(S\) has vertices at the points with coordinates \((0, 0)\), \((a, -a)\), \((2a, 0)\) and \((a, a)\) where \(a\) is a constant. The square \(S\) is transformed to the square \(S'\) by the transformation represented by \(\mathbf{M}\).

1. Determine, in terms of \(a\), the area of \(S'\) [2]

A transformation is equivalent to a shear parallel to the x-axis followed by a shear parallel to the y-axis and is represented by the matrix \(\begin{pmatrix} 1 & s \\ t & 0 \end{pmatrix}\). Find in terms of s the matrices which represent each of the shears. [7]

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 0.6 & 2.4 \\ -0.8 & 1.8 \end{pmatrix}\).

1. Find \(\det \mathbf{A}\). [1]

The matrix \(\mathbf{A}\) represents a stretch parallel to one of the coordinate axes followed by a rotation about the origin.

1. By considering the determinants of these transformations, determine the scale factor of the stretch. [2]
2. Explain whether the stretch is parallel to the \(x\)-axis or the \(y\)-axis, justifying your answer. [1]
3. Find the angle of rotation. [2]