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Transformation matrix problem

  • Thread starter Gregg
  • Start date
  • #1
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Homework Statement



7. (a) A transformation, T1 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: 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

[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.
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,832
249
[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.
Hi Gregg! :smile:

(very nice LaTeX, btw! :wink:)

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) :wink:
 
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