Geometric Transformations and Properties
Learning Targets: I can use matrices to perform geometric transformations. I can use matrices to calculate geometric properties of certain shapes.
Warm-Up
What transformation maps triangle ABC onto triangle (ABC)'?
First things first...
![Image](https://stage.geogebra.org/resource/txpsenk7/EimNfYmk7B6bdT3l/material-txpsenk7.png)
Reflections
Check out the 3 transformation matrices below. Use the GeoGebra applet below to explore each of these reflections by changing the elements in matrix A.
![Image](https://stage.geogebra.org/resource/g292zp95/bSasD7AsOWUT1CGv/material-g292zp95.png)
What is the reflection indicated by the matrix ?
Deformations
Check out some transformation matrices below, then use the GeoGebra applet to better understand each type of deformation by editing the elements in matrix A.
![Image](https://stage.geogebra.org/resource/xjahdddc/zJOc6ifBG0mEDSW7/material-xjahdddc.png)
Rotations (2-D)
![Image](https://stage.geogebra.org/resource/bwparagv/4WWjWOISm3arHkTC/material-bwparagv.png)
The matrix above is the 2-dimensional rotation matrix, which rotates CCW about the origin. See a demonstration in the applet below.
Do you think the determinant of the rotation matrix is dependent on the angle of rotation?
Rotations (3-D)
![Image](https://stage.geogebra.org/resource/uzwkz78k/8QqxbIXG8INkfREI/material-uzwkz78k.png)
We can rotate vectors in 3-dimensions too! Look at the 3 rotation matrices above. Each matrix rotates a vector CCW about a specific axis. Which one do you think rotates around the z-axis?
Check out the applet below.
Triangle Area
We can use matrices to calculate the area of a triangle given the Cartesian coordinates of the 3 vertices.
![Image](https://stage.geogebra.org/resource/gjy8dfxp/QsavKZBOIHYI8Oia/material-gjy8dfxp.png)
![Image](https://stage.geogebra.org/resource/jhxfe28q/oH4IMuUnkaWTofK2/material-jhxfe28q.png)
Tetrahedral Volume
We can use matrices to calculate the volume of a tetrahedron given the Cartesian coordinates of the 4 vertices.
![Image](https://stage.geogebra.org/resource/ekhvvr7v/YdVqNAG055EVGIZp/material-ekhvvr7v.png)
![Image](https://stage.geogebra.org/resource/au6xrut7/qGG0kQ2KIsokb5rf/material-au6xrut7.png)
Summary
Grab a PDF copy of the summary below.