How Do Matrix Transformations Alter 3D Space Geometrically?

[Gregg]

Homework Statement

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

r=\left(<br /> \begin{array}{c}<br /> x \\<br /> y \\<br /> z<br /> \end{array}<br /> \right)

r&#039;=\left(<br /> \begin{array}{c}<br /> x&#039; \\<br /> y&#039; \\<br /> z&#039;<br /> \end{array}<br /> \right)

and

<br /> M=\left(<br /> \begin{array}{ccc}<br /> 1 &amp; 0 &amp; 0 \\<br /> 0 &amp; 0 &amp; -1 \\<br /> 0 &amp; 1 &amp; 0<br /> \end{array}<br /> \right)

Describe the transformation geometrically.

(b)

Two other transformations are defined as follows: T<sub>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

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

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

Rotation about the x-axis 90 degrees.

(b)

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

T_2 =\left(<br /> \begin{array}{ccc}<br /> 1 &amp; 0 &amp; 0 \\<br /> 0 &amp; -1 &amp; 0 \\<br /> 0 &amp; 0 &amp; 1<br /> \end{array}<br /> \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.</sub>

 

[tiny-tim]

T=\left(<br /> \begin{array}{ccc}<br /> 1 &amp; 0 &amp; 0 \\<br /> 0 &amp; \text{cos$\theta $} &amp; -\text{sin$\theta $} \\<br /> 0 &amp; \text{sin$\theta $} &amp; \text{cos$\theta $}<br /> \end{array}<br /> \right)

Rotation about the x-axis 90 degrees.

(b)

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

T_2 =\left(<br /> \begin{array}{ccc}<br /> 1 &amp; 0 &amp; 0 \\<br /> 0 &amp; -1 &amp; 0 \\<br /> 0 &amp; 0 &amp; 1<br /> \end{array}<br /> \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.

Hi Gregg!

(very nice LaTeX, btw! )

your (a) is right.

your (b) is a reflection, but about the wrong axis

for (b) part2, the line x=0, y+z=0 is in the y-z plane (so not x = y + x)