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

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

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

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: 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

$\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \text{cos\theta } & -\text{sin\theta } \\ 0 & \text{sin\theta } & \text{cos\theta } \end{array} \right),\left( \begin{array}{ccc} \text{cos\theta } & 0 & \text{sin\theta } \\ 0 & 1 & 0 \\ -\text{sin\theta } & 0 & \text{cos\theta } \end{array} \right),\left( \begin{array}{ccc} \text{cos\theta } & -\text{sin\theta } & 0 \\ \text{sin\theta } & \text{cos\theta } & 0 \\ 0 & 0 & 1 \end{array} \right).$ 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.</sub>

 

[tiny-tim]

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

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.

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)

 

[Architeuthis Dux]

The given transformation matrix represents a reflection about the yz-plane, followed by a 90 degree rotation about the x-axis. This can be seen by looking at the values in the matrix, where the first row remains unchanged, the second row becomes the third row, and the third row becomes the negative of the second row. This results in a mirror image of the original coordinates across the yz-plane, and then a rotation of 90 degrees about the x-axis.

For part (b), the first transformation, T2, is a reflection in the x-y plane, which can be represented by the matrix shown. The second transformation, T3, is a rotation of 180 degrees about the line x=0, y+z=0. This can be thought of as a rotation about the 3D line x=y+z, as you correctly mentioned. This rotation can be represented by the matrix shown in the relevant information section.

For part (c), the combined transformations T3T1 and T1T3 can be easily determined by multiplying the matrices together. T3T1 represents a rotation of 180 degrees about the x-axis, followed by a reflection about the yz-plane. This can be thought of as a mirroring of the object across the x-axis, and then rotating it 180 degrees. T1T3, on the other hand, represents a reflection about the yz-plane, followed by a rotation of 180 degrees about the x-axis. This can be thought of as rotating the object 180 degrees, and then mirroring it across the x-axis. Both of these transformations result in the same final image, which is a reflection across the yz-plane and a rotation of 180 degrees about the x-axis.

Geometrically, these transformations can be visualized as manipulating an object in 3D space. The first transformation, T1, results in a mirror image of the object across the yz-plane, followed by a rotation of 90 degrees about the x-axis. T2, on the other hand, simply reflects the object across the x-y plane. T3 rotates the object 180 degrees about the line x=y+z, while T3T1 and T1T3 both result in a final image that is rotated and mirrored across the yz-plane.