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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...
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´╗┐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.
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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.
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´╗┐Rotations (2-D)
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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)
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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.
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´╗┐Tetrahedral Volume We can use matrices to calculate the volume of a tetrahedron given the Cartesian coordinates of the 4 vertices.
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Summary Grab a PDF copy of the summary below.

GeometryAndMatrices