Q39. - Algebra 2

Question

FOOTPRINTS

For exercises 38-41, use the following information.

The combination of a reflection and a translation is called a glide reflection.

An example is a set of footprints.

39. Write two matrix operations that can be used to find the coordinates of point $C$

Step-by-Step Solution

Answer

Two matrix operations that can be used to find coordinates of point $C$ are

Matrix Multiplication and Matrix Addition.

Step 1 - Vertex matrix of a point

An ordered pair $\left(x,y\right)$ is represented by a column matrix $\left[\begin{array}{c}x\\ y\\ \end{array}\right]$ which is known as vertex matrix of an ordered pair.

Step 2 - Vertex matrix after reflection over the x - a x i s

Vertex matrix of new point after reflection over the x -axis is product matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\\ & \end{array}\right]$ with vertex matrix of that point as shown below

$\left[\begin{array}{cc}1& 0\\ 0& -1\\ & \end{array}\right]\times \left[\begin{array}{c}x\\ y\\ \end{array}\right]=\left[\begin{array}{c}x\\ -y\\ \end{array}\right]$

So multiplication of matrices is the first matrix operation to get point $C$

Step 3 - Vertex matrix after translation

If the matrix is translated to $x\text{'}$ units right or left then $x\text{'}$ is added or subtracted from first row respectively.

If the matrix is translated to $y\text{'}$ units up or down then $y\text{'}$ is added or subtracted from second row respectively.

Thus translation matrix is of the form $\left[\begin{array}{c}x\text{'}\\ y\text{'}\\ \end{array}\right]$

As, vertex matrix after translation is sum of vertex matrix with translated matrix as shown below

$\left[\begin{array}{c}x\\ -y\\ \end{array}\right]+\left[\begin{array}{c}x\text{'}\\ y\text{'}\\ \end{array}\right]=\left[\begin{array}{c}x+x\text{'}\\ -y+y\text{'}\\ \end{array}\right]$

So addition of matrices is the second matrix operation to get point $C$