Matrix Rotation of a Figure

Background

Later, we will see how to reflect and dilate a figure using matrices. This first lesson in this series will cover the first matrix transformation, matrix rotation.

Before we continue, it's important that we review how to represent a function with a matrix. We put the ordered pair vertically in the matrix. Say we're given the points, $(x,y)$, $(a,b)$, and $(1, 2)$. We would represent these points in matrix form like so:

$$\begin{bmatrix} x&a&1 \\ y&b&2 \\ \end{bmatrix}$$

As we can see, the coordinate points are vertically paired in the matrix.

We actually haven't seen a method for rotation of a figure yet, but after this lesson, we will be able to do function rotations! In essence, what we will do is multiply a given matrix by another matrix to get new coordinate points.

$90^{\circ}$ Counterclockwise Rotation

To rotate a graphed figure by $90^{\circ}$ counterclockwise, we multiply a given matrix by the following matrix: $\begin{bmatrix}0&-1\\1&0\end{bmatrix}$

$180^{\circ}$ Counterclockwise Rotation

To rotate a graphed figure by $180^{\circ}$ counterclockwise, we multiply a given matrix by the following matrix: $\begin{bmatrix}-1&0\\0&-1\end{bmatrix}$

$270^{\circ}$ Counterclockwise Rotation

To rotate a graphed figure by $270^{\circ}$ counterclockwise, we multiply a given matrix by the following matrix: $\begin{bmatrix}0&1\\-1&0\end{bmatrix}$

$360^{\circ}$ Counterclockwise Rotation

To rotate a graphed figure by $360^{\circ}$ counterclockwise, we multiply a given matrix by teh following matrix: $\begin{bmatrix}1&0\\0&1\end{bmatrix}$

You may notice that this is an identity matrix! That's because when we rotate a figure by $360^{\circ}$, we are essentially turning it in a full circle, so we get the coordinate points for the original function!

(Video) Matrix Rotations, Rotate 90 Degrees

by David Staublin

This video by David Staublin walks you through an example of rotating a figure by $90 ^{\circ}$.

Summary

We can represent figures with matrices. If we're given the triangle $ABC$, with points, $A (0,0)$, $B(5,0)$, and $C(5,2)$. The green triangle represents Triangle $ABC$.

We can represent this triangle with the following matrix, $A$: $\begin{bmatrix} 0&5&5 \\ 0&0&2 \\ \end{bmatrix}$

The first row represents the $x$ values, and the second row represents the $y$ values. The first column is the coordinate point $A$, the second column is the coordinate point $B$, and the third point is the coordinate point $C$.

When we rotate this figure, we'll go point by point. Point $A$ at the origin will stay where it is, since we rotate figures around the origin. So, $A'$ is $(0,0)$. Next, we'll look at point $B$, which becomes $B'$ at $(0,5)$. Our last point $C$ rotates to $C'$ at $(-2, 5)$. Our new triangle looks like:

Our new figure can be represented by the following matrix $C$: $\begin{bmatrix} 0&0&-2 \\ 0&5&5 \\ \end{bmatrix}$

Now, we want to find a matrix that we can multiply by Matrix $A$ to give us Matrix $C$. It looks like our $x$ and $y$ values switch, and the $y$ value is negated. With some deducing, we can figure out the matrix that will rotate a figure by $90 ^{\circ}$ should be:

$$\begin{bmatrix} 0&-1 \\ 1&0 \\ \end{bmatrix}$$

Matrix Rotation of a Figure

We have already learned how to rotate figures around a coordinate system. Wit

Before we can rotate a figure using matrices, we need to first learn how to arrange the coordinate points in a matrix. If we have two points, $(1,2)$ and $(2,3)$, we can arrange them in the following matrix:

$$\begin{bmatrix} 1&2 \\ 2&3 \\ \end{bmatrix}$$

If we want to rotate this matrix, we can multiply this matrix by one of the following matrices, depending on our desired rotation (the angle of rotation). These are the rotation matrices.

Let's go back to our previous points,

$$\begin{bmatrix} 1&2 \\ 2&3 \\ \end{bmatrix}$$

If we want to rotate this figure by $270 ^{\circ}$ counterclockwise, we can multiply:

$$\begin{bmatrix} 1&2 \\ 2&3 \\ \end{bmatrix} \cdot \begin{bmatrix} 0&1 \\ -1&0 \\ \end{bmatrix}$$

Use this same system to do other counterclockwise and clockwise rotations.

Looking Further

In the future, when learning linear algebra, you may use this technique with orthogonal matrices and in the euclidean space. Mastering this technique early will help you in the future!