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## Homework Statement

7. (a) A transformation, T

_{1}of three dimensional space is given by

**r'**=

**Mr**, where

[itex]r=\left(

\begin{array}{c}

x \\

y \\

z

\end{array}

\right)[/itex]

[itex]r'=\left(

\begin{array}{c}

x' \\

y' \\

z'

\end{array}

\right)[/itex]

and

[itex]

M=\left(

\begin{array}{ccc}

1 & 0 & 0 \\

0 & 0 & -1 \\

0 & 1 & 0

\end{array}

\right)[/itex]

Describe the transformation geometrically.

(b)

Two other transformations are defined as follows: T

_{2 is a reflection in the x-y plane, and 3 is a rotation through 180 degrees about the line x=0, y+z=0. By considering the image under each transformation of the points with position vectors, i,j,k or otherwise find a matrix for each T2/ (c) Determine the matrixes for the combined transformations of T3T1 amd T1T3 amd describe each of these tranformations geometrically. 2. Relevant information [itex]\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \text{cos$\theta $} & -\text{sin$\theta $} \\ 0 & \text{sin$\theta $} & \text{cos$\theta $} \end{array} \right),\left( \begin{array}{ccc} \text{cos$\theta $} & 0 & \text{sin$\theta $} \\ 0 & 1 & 0 \\ -\text{sin$\theta $} & 0 & \text{cos$\theta $} \end{array} \right),\left( \begin{array}{ccc} \text{cos$\theta $} & -\text{sin$\theta $} & 0 \\ \text{sin$\theta $} & \text{cos$\theta $} & 0 \\ 0 & 0 & 1 \end{array} \right).[/itex] represent rotations of theta degrees about the x-,y- and z-axes. 3. Attempt [itex]T=\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \text{cos$\theta $} & -\text{sin$\theta $} \\ 0 & \text{sin$\theta $} & \text{cos$\theta $} \end{array} \right)[/itex] Rotation about the x-axis 90 degrees. (b) [itex]T_2:{x,y,z} \to {x,-y,z} [/itex] [itex] T_2 =\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{array} \right)[/itex] (b) I am stuck here on how to do a rotation about the line x=0, y+z=0. Does this imply it is about the 3D line x=y+z. (c) This will be simple once I have done the other part.}