Reflection in the Line y = x Matrix

Reflection in the Line y = x Matrix

How to use the reflection in the line y = x matrix to find the image under the reflection: the matrix, example, and its solution.

Matrix

The reflection in the line y = x matrix is [0 1 / 1 0]. To find the coordinates of the image, multiply the reflection matrix in front of the vertex matrix.

The reflection in the line y = x matrix is
[1 0 / 0 -1].

To find the coordinates of the image,
multiply the reflection matrix [0 1 / 1 0]
in front of the vertex matrix.

Reflection in the line y = x

Multiplying matrices

Example

Triangle ABC has vertices A(6, 4), B(7, 1), and C(2, -2). Triangle A'B'C' is the image of triangle ABC under the reflection in the line y = x. Find the coordinates of the vertices of triangle A'B'C'.

Previously, you've solved this example.

Reflection in the line y = x

Let's solve the same example
by using the reflection in y = x matrix.

A(6, 4), B(7, 1), C(2, -2)

So the vertex matrix is
[6 7 2 / 4 1 -2].

The reflection in y = x matrix is
[0 1 / 1 0].

So multiply the reflection matrix [0 1 / 1 0]
in front of the vertex matrix [6 7 2 / 4 1 -2].

Multiply these two matrices.

Multiplying matrices

[Row 1]⋅[Column 1]: 0⋅6 + 1⋅4
[Row 1]⋅[Column 2]: 0⋅7 + 1⋅1
[Row 1]⋅[Column 3]: 0⋅2 + 1⋅(-2)

[Row 2]⋅[Column 1]: 1⋅6 + 0⋅4
[Row 2]⋅[Column 2]: 1⋅7 + 0⋅1
[Row 2]⋅[Column 3]: 1⋅2 + 0⋅(-2)

Then the vertex matrix of the image is
[4 1 -2 / 6 7 2].

The first column is [4 / 6].
So A'(4, 6).

The second column is [1 / 7].
So B'(1, 7).

The third column is [-2 / 2].
So C'(-2, 2).

These are the coordinates of the image.

Here's what you've solved.

By multiplying
the reflection in y = x matrix [0 1 / 1 0],
you found the vertices of the image (△A'B'C')
under the reflection in y = x.