Exercise 2 Page 806

We want to use matrices to perform a rotation by 270^(∘) about the origin on the triangle with vertices at A(1,1), B(2,4), and C(4,- 1). Let's start by drawing the triangle on the coordinate plane.

A rotation turns a figure about a fixed point called the center of rotation. We can multiply a figure's vertex matrix by a rotation matrix to find the vertices of the rotated image. To create the vertex matrix of a figure, we write each vertex as the column of a matrix.

A B C ↓ ↓ ↓ 1 & 2 & 4 1 & 4 & - 1 Then, we multiply the corresponding rotation matrix by the obtained matrix. Let's see the different rotation matrices for the coordinate plane.

90^(∘) Rotation Matrix	180^(∘) Rotation Matrix	270^(∘) Rotation Matrix	360^(∘) Rotation Matrix
0 & - 1 1 & 0	- 1 & 0 0 & - 1	0 & 1 - 1 & 0	1 & 0 0 & 1

The multiplication will result in the image matrix. Here, each column contains the coordinates of the vertices of the image of △ABC after the rotation.

0 & 1 - 1 & 0 1 & 2 & 4 1 & 4 & - 1

MultMatrices

Multiply matrices

0(1)+1(1) & 0(2)+1(4) & 0(4)+1(- 1) - 1(1)+0(1) & -1 (2)+0(4) & - 1(4)+0(- 1)

Multiply

Multiply

0+1 & 0+4 & 0+(- 1) - 1+0 & - 2+0 & - 4+0

AddTerms

Add terms

1 & 4 & - 1 - 1 & - 2 & - 4

The image of △ABC after a 270^(∘) rotation is △A'B'C' with vertices at A'(1,- 1), B'(4,- 2), and C'(- 1,- 4).