Reflecting a Triangle Across a Non-Standard Line Using Matrix Transformations

Matrix transformations are a powerful tool for performing geometric transformations, such as reflecting a triangle across a given line. This article will walk you through the process of finding the reflection of a triangle defined by the vertices A (2, 3), B (-2, -3), and C (5, 3) under the mirror line y 3x - 1. Let's explore the matrix transformation process step-by-step.

Step 1: Finding the Unit Normal Vector

The first step is to find the unit normal vector of the line y 3x - 1. This line can be represented in the general form as 3x - y - 1 0. The coefficients of this equation, A 3, B -1, and C -1, can be used to find the normal vector. The normal vector is given by:

n (A, B) (3, -1)

To find the unit normal vector, we need to normalize the normal vector by dividing it by its magnitude:

length sqrt{A^2 B^2} sqrt{3^2 (-1)^2} sqrt{10}

The unit normal vector is:

u

Step 2: Constructing the Reflection Matrix

Next, we construct the reflection matrix M. This matrix is given by:

M I - 2uu^T

where I is the identity matrix and uu^T is the outer product of u. The outer product is calculated as:

uu^T begin{pmatrix} frac{3}{sqrt{10}} frac{-1}{sqrt{10}} end{pmatrix} begin{pmatrix} frac{3}{sqrt{10}} frac{-1}{sqrt{10}} end{pmatrix} begin{pmatrix} frac{9}{10} frac{-3}{10} frac{-3}{10} frac{1}{10} end{pmatrix}

The reflection matrix M can now be calculated as:

M I - 2 begin{pmatrix} frac{9}{10} frac{-3}{10} frac{-3}{10} frac{1}{10} end{pmatrix}

M begin{pmatrix} 1 0 0 1 end{pmatrix} - 2 begin{pmatrix} frac{9}{10} frac{-3}{10} frac{-3}{10} frac{1}{10} end{pmatrix}

M begin{pmatrix} 1 0 0 1 end{pmatrix} - begin{pmatrix} frac{18}{10} frac{-6}{10} frac{-6}{10} frac{2}{10} end{pmatrix}

M begin{pmatrix} 1 - frac{18}{10} 0 frac{6}{10} 0 1 - frac{2}{10} frac{6}{10} end{pmatrix}

M begin{pmatrix} -frac{8}{10} frac{6}{10} frac{6}{10} -frac{8}{10} end{pmatrix}

Step 3: Applying the Reflection Matrix to Each Vertex

With the reflection matrix M in hand, we can now apply it to each vertex of the triangle to find the reflected vertices. Let's start with vertex A (2, 3):

A' M begin{pmatrix} 2 3 end{pmatrix} begin{pmatrix} -frac{8}{10} frac{6}{10} frac{6}{10} -frac{8}{10} end{pmatrix} begin{pmatrix} 2 3 end{pmatrix}

A' begin{pmatrix} -frac{16}{10} frac{18}{10} frac{12}{10} - frac{24}{10} end{pmatrix} begin{pmatrix} 0.2 -1.2 end{pmatrix} (0.4, 2.2)

Next, for vertex B (-2, -3):

B' M begin{pmatrix} -2 -3 end{pmatrix} begin{pmatrix} -frac{8}{10} frac{6}{10} frac{6}{10} -frac{8}{10} end{pmatrix} begin{pmatrix} -2 -3 end{pmatrix}

B' begin{pmatrix} frac{16}{10} frac{18}{10} -frac{12}{10} frac{24}{10} end{pmatrix} begin{pmatrix} 3.4 -3.6 end{pmatrix}

Finally, for vertex C (5, 3):

C' M begin{pmatrix} 5 3 end{pmatrix} begin{pmatrix} -frac{8}{10} frac{6}{10} frac{6}{10} -frac{8}{10} end{pmatrix} begin{pmatrix} 5 3 end{pmatrix}

C' begin{pmatrix} -4 1.8 3 - 2.4 end{pmatrix} begin{pmatrix} -2.2 5.4 end{pmatrix}

The final reflected vertices of the triangle are:

A' (0.4, 2.2) B' (3.4, -3.6) C' (-2.2, 5.4)

Thus, the reflection of the triangle across the line y 3x - 1 is achieved with the vertices A' (0.4, 2.2), B' (3.4, -3.6), and C' (-2.2, 5.4).

Conclusion

Matrix transformations, especially when carefully applied, provide a robust framework for performing various geometric transformations. This method can be extended to other lines and shapes, making it a valuable tool in both theoretical and applied mathematics.