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Lecture 3

MA 108 Lecture Notes - Lecture 3: Rotation Matrix, Main Diagonal, Identity Matrix


Department
Mathematics
Course Code
MA 108
Professor
Elyse Suzanne Rogers
Lecture
3

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4) Rotation: In a rotation, all points in the preimage turn in the same direction about a fixed
point (the center of rotation) through a fixed angle (the angle of rotation). If a point A is
rotated about a center C, then the angle ACA’ is the angle of rotation. The direction of the
rotation is assumed to be counterclockwise, unless otherwise stated.
A rotation is an isometry, so rotations preserve size (and shape). Rotations also
preserve the orientation of figures.
While rotation is a quite general concept and the center and angle of rotation can,
respectively, be anywhere in the plane and can take on any measure, we will concentrate on
four specific types of rotation, all counterclockwise with center at the origin: 90 degrees, 180
degrees, 270 degrees, and 360 degrees.
Case 1: 90 degree rotation about the origin. When a point is rotated 90 degrees, the x-
coordinate of the image is the opposite of the original y-coordinate, and the y-coordinate of the
image is the original x-coordinate. In coordinate notation, (x’, y’) = ( −y, x ). For example, the
point ( 3, 5) has the image ( -5, 3), and the point ( -2, -7) has the image ( 7, -2).
If a figure is represented as a polygon matrix, then it can be rotated 90 degrees by
multiplying the polygon matrix by the rotation matrix [ 0 -1 . The rotation matrix goes in front
1 0 ]
of the polygon matrix (this is called “premultiplying”). For example, suppose we want to rotate
triangle ABC 90 degrees, where A = ( 1, 6), B = ( 3, 8), and C = ( 5, 4). The matrix multiplication
and image look like this:
[ 0 -1 [ 1 3 5 = [ -6 -8 -4
1 0 ] 6 8 4 ] 1 3 5 ]
Try these: Find the image of each point after a 90-degree rotation:
1) ( 9, 4) ( -4, 9 )
2) (-4, 7) ( -7, -4)
3) ( 2, -5) ( 5, 2 )
4) ( -5, -3) ( 3, -5)
Write the matrix equation to rotate triangle ABC [A = ( 0, -2), B = ( 6, 1), C = ( 4, 3)] 90 degrees.
[ 0 -1 [ 0 6 4 = [ 2 -1 -3
1 0 ] -2 1 3 ] 0 6 4 ]
Case 2: 180 degree rotation about the origin. When a point is rotated 180 degrees, the x-
and y-coordinates of the image are the opposites of the original x- and y-coordinates. In other
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