Using Matrices to Model Translations

Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson will demonstrate how to use matrices to find translations of images in the coordinate plane. Coordinate matrices, the addition operation of matrices, and translations will be defined and examples are provided.

Translations

Have you ever graphed a rectangle in the coordinate plane? Does it matter where the vertices of the rectangle are? If we change the coordinates of the rectangle, is it still the same rectangle? We can take a rectangle and graph it in the coordinate plane by plotting the vertices. If we shift the rectangle to the left or right and up or down, we will still have the same size rectangle. The only thing that changes when we shift it is the location of the vertices.

A translation is a change in the location of all the points of an object in the coordinate plane. We can translate objects to the left or right by adding or subtracting from the x-values. We can translate an object up or down by adding or subtracting from the y-values.

Suppose we have the rectangle with vertices at A(0,0), B(0, 3), C(4, 3) and D(4, 0), and we want to translate three units to the left and two units up. We are going to find the coordinates of the new image, A'B'C'D' by using a matrix.

Coordinate and Translation Matrices

A matrix is a rectangular array of numbers that represents something. It is a way to organize data. First, we will place the coordinates in a matrix. This is called a coordinate matrix.

The x values are written in the first row and the y values are written in the second row. Each point represents a different column.

Now we will write the translation matrix. This matrix will be the same size as the coordinate matrix, having four columns and two rows. Since the x-values represent the first row, all four columns will have -3 in the first row. This is because we need to subtract 3 from each x-value to move the rectangle to the left. The y-values represent the second row, so all four columns will have a 2. This is because we need to add two to all the y-values to shift the rectangle up two units.

To perform the translation, we simply add the matrices together. In order to add, we add the corresponding elements in each matrix.

The resulting matrix has the coordinates of the new image.

We can graph both rectangles in the coordinate plane.

Do you notice that both rectangles are the same? The only difference is the location of the vertices. The green rectangle has been moved three units to the left and two units up.

Another Example

Now, let's look at another object. Suppose you have the triangle with vertices A(2, 1), B(2, 6) and C(6, 1). Use matrices to find the coordinates of the image if we want to translate the triangle 7 units to the left and 8 units down.

First, we can put the coordinates of the triangle in a coordinate matrix.

Second, we can write the translation rule in a matrix. The first row will be -7 because we need to subtract 7 from each x-value to move the triangle to the left. The second row will be -8 because we need to subtract 8 from all the y-values to move the triangle down.

Next, we will add the two matrices together to get our new coordinates.

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