# Transformation matrix problem

1. May 24, 2009

### Gregg

1. The problem statement, all variables and given/known data

7. (a) A transformation, T1 of three dimensional space is given by r'=Mr, where

$r=\left( \begin{array}{c} x \\ y \\ z \end{array} \right)$

$r'=\left( \begin{array}{c} x' \\ y' \\ z' \end{array} \right)$

and

$M=\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{array} \right)$

Describe the transformation geometrically.

(b)

Two other transformations are defined as follows: T2 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

$\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \text{cos\theta } & -\text{sin\theta } \\ 0 & \text{sin\theta } & \text{cos\theta } \end{array} \right),\left( \begin{array}{ccc} \text{cos\theta } & 0 & \text{sin\theta } \\ 0 & 1 & 0 \\ -\text{sin\theta } & 0 & \text{cos\theta } \end{array} \right),\left( \begin{array}{ccc} \text{cos\theta } & -\text{sin\theta } & 0 \\ \text{sin\theta } & \text{cos\theta } & 0 \\ 0 & 0 & 1 \end{array} \right).$ represent rotations of theta degrees about the x-,y- and z-axes.

3. Attempt
$T=\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \text{cos\theta } & -\text{sin\theta } \\ 0 & \text{sin\theta } & \text{cos\theta } \end{array} \right)$

Rotation about the x-axis 90 degrees.

(b)

$T_2:{x,y,z} \to {x,-y,z}$

$T_2 =\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{array} \right)$

(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.

2. May 24, 2009

### tiny-tim

Hi Gregg!

(very nice LaTeX, btw! )