Question

suppose p is the line with equation y = 5 and q is the line with equation y = 6. write the following composition of reflections as one translation. rp ∘ rq rp ∘ rq = t

Explanation:

Step1: Recall reflection - translation relationship

The composition of two reflections $R_p$ and $R_q$ over two parallel lines $p:y = 5$ and $q:y=6$ is a translation. The distance between the two parallel lines $d=\vert6 - 5\vert=1$. When we have a composition of reflections $R_p\circ R_q$ over two parallel lines, the translation vector is perpendicular to the lines and its magnitude is $2d$.

Step2: Determine the translation vector

Since the lines are horizontal ($y -$ constant), the translation is in the vertical direction. The distance between the lines $y = 5$ and $y = 6$ is $1$. The composition $R_p\circ R_q$ is a translation by a vector $\vec{v}=(0, - 2)$. The negative sign indicates the direction from the line $y = 6$ towards $y = 5$ (because of the order of the composition $R_p\circ R_q$ which means we first reflect over $q$ and then over $p$).

Answer:

A translation by the vector $(0,-2)$