Reflection in the Origin Matrix

Reflection in the Origin Matrix

How to use the reflection in the origin matrix to find the image under the reflection: the matrix, example, and its solution.

Matrix

The reflection in the origin matrix is [-1 0 / 0 -1], which is -I. To find the coordinates of the image, multiply the reflection matrix in front of the vertex matrix.

The reflection in the origin matrix is
[-1 0 / 0 -1],
which is -I.

(I is the identity matrix.)

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

Reflection in the origin

Multiplying matrices

Example

Triangle ABC has vertices A(-3, 5), B(5, 4), and C(2, -1). Triangle A'B'C' is the image of triangle ABC under the reflection in the origin. Find the coordinates of the vertices of triangle A'B'C'.

Previously, you've solved this example.

Reflection in the origin

Let's solve the same example
by using the reflection in the origin matrix.

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

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

The reflection in the origin matrix is
[-1 0 / 0 -1].

So multiply the reflection matrix [-1 0 / 0 -1]
in front of the vertex matrix [-3 5 2 / 5 4 -1].

Multiply these two matrices.

Multiplying matrices

[Row 1]⋅[Column 1]: (-1)⋅(-3) + 0⋅5
[Row 1]⋅[Column 2]: (-1)⋅5 + 0⋅4
[Row 1]⋅[Column 3]: (-1)⋅2 + 0⋅(-1)

[Row 2]⋅[Column 1]: 0⋅(-3) + (-1)⋅5
[Row 2]⋅[Column 2]: 0⋅5 + (-1)⋅4
[Row 2]⋅[Column 3]: 0⋅2 + (-1)⋅(-1)

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

The first column is [3 / -5].
So A'(3, -5).

The second column is [-5 / -4].
So B'(-5, -4).

The third column is [-2 / 1].
So C'(-2, 1).

These are the coordinates of the image.

Here's what you've solved.

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