Reflection Matrix

Reflection Matrix for Horizontal Axis

Recall that reflecting a point $(x,y)$ across the $x$-axis results in the image $(x,-y)$.

We are looking for a matrix $\begin{pmatrix} r & s \\ t & u \end{pmatrix}$ such that:

By equating the coefficients, we get:

• $rx + sy = 1x + 0y \implies r=1, s=0$
• $tx + uy = 0x - 1y \implies t=0, u=-1$

Thus, the reflection matrix for the $x$-axis is $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$.

Reflection Matrix for Vertical Axis

Reflecting a point $(x,y)$ across the $y$-axis results in $(-x,y)$.

This gives $r=-1, s=0, t=0, u=1$.

The matrix is $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$.

Reflection Matrix for Main Diagonal Line

Reflecting a point $(x,y)$ across the line $y=x$ results in $(y,x)$.

This gives $r=0, s=1, t=1, u=0$.

The matrix is $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$.

Reflection Matrix for Negative Diagonal Line

Reflecting a point $(x,y)$ across the line $y=-x$ results in $(-y,-x)$.

This gives $r=0, s=-1, t=-1, u=0$.

The matrix is $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$.

Basic Reflection Matrices

• Reflection matrix for the $x$-axis: $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
• Reflection matrix for the $y$-axis: $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$
• Reflection matrix for line $y=x$: $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
• Reflection matrix for line $y=-x$: $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$

Finding the Image of a Point using Matrix

Find the image of $(3,-4)$ reflected across the $x$-axis.

Alternative Solution:

Using the reflection matrix for the $x$-axis:

Thus, the image is $(3,4)$.

Finding the Image of a Triangle using Matrix

Determine the image of triangle $ABC$ with vertices $A(3,1)$, $B(-2,3)$, and $C(2,-1)$ reflected across the $y$-axis!

Alternative Solution:

The reflection matrix for the $y$-axis is $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$.

The matrix of triangle $ABC$'s vertices: $\begin{pmatrix} 3 & -2 & 2 \\ 1 & 3 & -1 \end{pmatrix}$.

Thus, the image is triangle $A'B'C'$ with vertices $A'(-3,1)$, $B'(2,3)$, and $C'(-2,-1)$.

Exercises

1. Find the image of point $(3,5)$ reflected across the $x$-axis using the matrix multiplication $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$.
2. Determine the image of triangle $ABC$ with vertices $A(3,1)$, $B(-2,3)$, and $C(2,-1)$ reflected across the line $y=-x$!

Key Answers

1. The image of point $(3,5)$ reflected across the $x$-axis using the matrix multiplication $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ is:

   $$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 3 \\ 5 \end{pmatrix} = \begin{pmatrix} 3 \\ -5 \end{pmatrix}$$

   Image: $(3,-5)$.

2. The reflection matrix for $y=-x$ is $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$.

   Matrix of ABC vertices: $\begin{pmatrix} 3 & -2 & 2 \\ 1 & 3 & -1 \end{pmatrix}$.

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

   $$= \begin{pmatrix} (0)(3)+(-1)(1) & (0)(-2)+(-1)(3) & (0)(2)+(-1)(-1) \\ (-1)(3)+(0)(1) & (-1)(-2)+(0)(3) & (-1)(2)+(0)(-1) \end{pmatrix}$$

   $$= \begin{pmatrix} -1 & -3 & 1 \\ -3 & 2 & -2 \end{pmatrix}$$

   Image: $A'(-1,-3)$, $B'(-3,2)$, $C'(1,-2)$.