Rotation Matrix: 90 Degrees Counterclockwise

How to use the rotation 90 degrees counterclockwise matrix to find the image under the rotation: formula, 1 example, and its solution.

Formula

Formula

The rotation 90º counterclockwise matrix is
[0 -1 / 1 0].

Example

Example

Solution

The image is under
the rotation of 90º counterclockwise
about the origin.

So write the rotation matrix
[0 -1 / 1 0].

Write the vertex matrix.

A(2, 1), B(3, 4), C(5, 3)

So the vertex matrix is
[2 3 5 / 1 4 3].

So the vertex matrix of the image is
[0 -1 / 1 0][2 3 5 / 1 4 3].

Solve [0 -1 / 1 0][2 3 5 / 1 4 3].

Multiply Matrices

Row 1, column 1:
0⋅2 + (-1)⋅1

Row 1, column 2:
0⋅3 + (-1)⋅4

Row 1, column 3:
0⋅5 + (-1)⋅3

Row 2, column 1:
1⋅2 + 0⋅1

Row 2, column 2:
1⋅3 + 0⋅4

Row 2, column 3:
1⋅5 + 0⋅3

This is the vertex matrix of the image.

0⋅2 + (-1)⋅1
= 0 - 1

0⋅3 + (-1)⋅4
= 0 - 4

0⋅5 + (-1)⋅3
= 0 - 3

1⋅2 + 0⋅1
= 2 + 0

1⋅3 + 0⋅4
= 3 + 0

1⋅5 + 0⋅3
= 5 + 0

0 - 1 = -1

0 - 4 = -4

0 - 3 = -3

2 + 0 = 2

3 + 0 = 3

5 + 0 = 5

[-1 -4 -3 / 2 3 5]
is the vertex matrix of the image.

So column 1 is the image of A:
A'(-1, 2).

Column 2 is the image of B:
B'(-4, 3).

Column 3 is the image of C:
C'(-3, 5).

So
A'(-1, 2)
B'(-4, 3)
C'(-3, 5)
is the answer.

Graph

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

The image is under
the rotation of 90º counterclockwise
about the origin.