Rotation Matrix

How to use the rotation matrix to find the image under a rotation (angle = theta): formula, 1 example, and its solution.

Formula

Formula

For the rotation of an angle θ
counterclockwise about the origin,
the rotation matrix is
[cos θ -sin θ / sin θ cos θ].

To use the rotation matrix,
you should know
how to find the trigonometric function values
sine and cosine.

Example

Example

Solution

The image is under
the rotation of 60º counterclockwise
about the origin.

So write the rotation matrix
[cos 60º -sin 60º / sin 60º cos 60º].

Write the vertex matrix.

A(2, 2), B(4, 2), C(4, 4)

So the vertex matrix is
[2 4 4 / 2 2 4].

So the vertex matrix of the image is
[cos 60º -sin 60º / sin 60º cos 60º][2 4 4 / 2 2 4].

To find sin 60º and cos 60º,

draw a 30-60-90 triangle
whose sides are 1, √3, 2.

cos 60º

Cosine is CAH:
Cosine,
Adjacent side (1),
Hypotenuse (2).

So cos 60º = 1/2.

sin 60º

Sine is SOH:
Sine,
Opposite side (√3),
Hypotenuse (2).

So sin 60º = √3/2.

So -sin 60º = -√3/2.

sin 60º = √3/2

cos 60º = 1/2

Write the vertex matrix.

So
[cos 60º -sin 60º / sin 60º cos 60º][2 4 4 / 2 2 4]
= [1/2 -√3/2 / √3/2 1/2][2 4 4 / 2 2 4].

[1/2 -√3/2 / √3/2 1/2]
= (1/2)[1 -√3 / √3 1]

[2 4 4 / 2 2 4]
= 2[1 2 2 / 1 1 2]

Cancel the coefficients (1/2) and 2.

Solve [1 -√3 / √3 1][1 2 2 / 1 1 2].

Multiply Matrices

Row 1, column 1:
1⋅1 + (-√3)⋅1

Row 1, column 2:
1⋅2 + (-√3)⋅1

Row 1, column 3:
1⋅2 + (-√3)⋅2

Row 2, column 1:
3⋅1 + 1⋅1

Row 2, column 2:
3⋅2 + 1⋅1

Row 2, column 3:
3⋅2 + 1⋅2

This is the vertex matrix of the image.

1⋅1 + (-√3)⋅1
= 1 - √3

1⋅2 + (-√3)⋅1
= 2 - √3

1⋅2 + (-√3)⋅2
= 2 - 2√3

3⋅1 + 1⋅1
= √3 + 1

3⋅2 + 1⋅1
= 2√3 + 1

3⋅2 + 1⋅2
= 2√3⋅2 + 2

This is the vertex matrix of the image.

So column 1 is the image of A:
A'(1 - √3, 1 + √3).

Column 2 is the image of B:
B'(2 - √3, 1 + 2√3).

Column 3 is the image of C:
C'(2 - 2√3, 2 + 2√3).

So
A'(1 - √3, 1 + √3)
B'(2 - √3, 1 + 2√3)
C'(2 - 2√3, 2 + 2√3)
is the answer.

Graph

This is the graph of △ABC
and its image △A'B'C'.

The image is under
the rotation of 60º counterclockwise
about the origin.