Q40. - 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.

40. Does it matter which operation you do first? Explain.

Step-by-Step Solution

Answer

No, it does not matter which matter will be operated first.

Multiplication of matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\\ & \end{array}\right]$ with vertex matrix of point $A$, then addition of translated matrix $\left[\begin{array}{c}6\\ 0\\ \end{array}\right]$ gives vertex matrix of point $B$.

Also addition of translated matrix $\left[\begin{array}{c}6\\ 0\\ \end{array}\right]$ with vertex matrix of A, then multiplication of it with matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\\ & \end{array}\right]$ gives vertex matrix of point B which shows order does not matter.

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.

For the coordinates $A\left(5,-2\right)$ and $B\left(11,2\right)$, vertex matrices are $\left[\begin{array}{c}5\\ -2\\ \end{array}\right]\text{ and }\left[\begin{array}{c}11\\ 2\\ \end{array}\right]$ respectively.

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

Multiply matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\\ & \end{array}\right]$ with vertex matrix of A, then add translated matrix $\left[\begin{array}{c}6\\ 0\\ \end{array}\right]$ as shown below

 $\begin{array}{c}\left(\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\times \left[\begin{array}{c}5\\ -2\end{array}\right]\right)+\left[\begin{array}{c}6\\ 0\end{array}\right]=\left[\begin{array}{c}5\\ 2\end{array}\right]+\left[\begin{array}{c}6\\ 0\end{array}\right]\\ =\left[\begin{array}{c}5+6\\ 2+0\end{array}\right]\\ =\left[\begin{array}{c}11\\ 2\end{array}\right]\end{array}$

So vertex matrix after reflection over x-axis followed by translation of 6 units to right gives vertex matrix of coordinate B

Step 3 - Vertex matrix after translation followed by reflection over the x - a x i s

Add translated matrix $\left[\begin{array}{c}6\\ 0\\ \end{array}\right]$ with vertex matrix of A, then multiply it with matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\\ & \end{array}\right]$ as shown below

$\begin{array}{c}\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\times \left(\left[\begin{array}{c}5\\ -2\end{array}\right]+\left[\begin{array}{c}6\\ 0\end{array}\right]\right)=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\times \left[\begin{array}{c}5+6\\ -2+0\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\times \left[\begin{array}{c}11\\ -2\end{array}\right]\\ =\left[\begin{array}{c}11\\ 2\end{array}\right]\end{array}$ 

So vertex matrix after translation of 6 units to the right followed by reflection over x-axis gives vertex matrix of coordinate B

Thus, it does not matter which matter will be operated first.