3D transformation matrices

8. The point \(( x , y , z )\) is rotated through \(60 ^ { \circ }\) anticlockwise around the \(z\)-axis. After rotation, the value of the \(x\)-coordinate is equal to the value of the \(y\)-coordinate.
Show that \(y = ( a + \sqrt { b } ) x\), where \(a\), \(b\) are integers whose values are to be determined.

4 You are given that the matrix \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & \frac { 2 a - a ^ { 2 } } { 3 } & 0 \\ 0 & 0 & 1 \end{array} \right)\), where \(a\) is a positive constant, represents the transformation R which is a reflection in 3-D.

1. State the plane of reflection of R .
2. Determine the value of \(a\).
3. With reference to R explain why \(\mathbf { A } ^ { 2 } = \mathbf { I }\), the \(3 \times 3\) identity matrix.

6 Anna has been asked to describe the transformation given by the matrix $$\left[ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & - \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\ 0 & \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right]$$ She writes her answer as follows: The transformation is a rotation about the \(x\)-axis through an angle of \(\theta\), where $$\begin{gathered} \sin \theta = \frac { 1 } { 2 } \quad \text { and } \quad - \sin \theta = - \frac { 1 } { 2 } \\ \theta = 30 ^ { \circ } \end{gathered}$$ Identify and correct the error in Anna's work.
[0pt] [2 marks] \(7 \quad\) Prove by induction that, for all integers \(n \geq 1\), the expression \(7 ^ { n } - 3 ^ { n }\) is divisible by 4

3 The matrix \(\mathbf { M }\) represents a rotation about the \(x\)-axis. $$\mathbf { M } = \left[ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & a & \frac { \sqrt { 3 } } { 2 } \\ 0 & b & - \frac { 1 } { 2 } \end{array} \right]$$ Which of the following pairs of values is correct?
Tick ( \(\checkmark\) ) one box. $$\begin{array} { l l } a = \frac { 1 } { 2 } \text { and } b = \frac { \sqrt { 3 } } { 2 } & \square \\ a = \frac { 1 } { 2 } \text { and } b = - \frac { \sqrt { 3 } } { 2 } & \square \\ a = - \frac { 1 } { 2 } \text { and } b = \frac { \sqrt { 3 } } { 2 } & \square \\ a = - \frac { 1 } { 2 } \text { and } b = - \frac { \sqrt { 3 } } { 2 } & \end{array}$$

3 Which of the following transformations is represented by the matrix \(\left[ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 1 \end{array} \right]\) ?
Tick ( \(\checkmark\) ) one box.
[0pt] [1 mark] Rotation of \(180 ^ { \circ }\) about the \(x\)-axis □ Reflection in the plane \(x = 0\) □ Rotation of \(180 ^ { \circ }\) about the \(y\)-axis □ Reflection in the plane \(y = 0\) □

7 Three non-singular square matrices, A, B and \(\mathbf { R }\) are such that $$A R = B$$ The matrix \(\mathbf { R }\) represents a rotation about the \(z\)-axis through an angle \(\theta\) and $$\mathbf { B } = \left[ \begin{array} { c c c } - \cos \theta & \sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{array} \right]$$ 7

1. Show that \(\mathbf { A }\) is independent of the value of \(\theta\).
   7
2. Give a full description of the single transformation represented by the matrix \(\mathbf { A }\).

2 Which one of the matrices below represents a rotation of \(90 ^ { \circ }\) about the \(x\)-axis? Circle your answer.
[0pt] [1 mark] \(\left[ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 1 \end{array} \right]\) \(\left[ \begin{array} { c c c } - 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]\) \(\left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array} \right]\) \(\left[ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & 0 \end{array} \right]\)

2
You are given the matrix \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & - 1 & 0 \end{array} \right)\).

1. Find \(\mathbf { A } ^ { 4 }\).
2. Describe the transformation that A represents. The matrix \(\mathbf { B }\) represents a reflection in the plane \(x = 0\).
3. Write down the matrix \(B\). The point \(P\) has coordinates \(( 2,3,4 )\). The point \(P ^ { \prime }\) is the image of \(P\) under the transformation represented by \(\mathbf { B }\).
4. Find the coordinates of \(P ^ { \prime }\).