Question

suppose k is the line with equation x = - 2 and l is the line with equation x = - 1. write the following composition of reflections as one translation. rₖ ∘ rₗ rₖ ∘ rₗ=t

Explanation:

Step1: Recall the rule for composition of reflections

The composition of two reflections $R_{l_1}\circ R_{l_2}$ over parallel lines $l_1$ and $l_2$ is a translation. If the distance between two parallel lines $l_1:x = a_1$ and $l_2:x=a_2$ is $d=\vert a_1 - a_2\vert$, the translation vector for $R_{l_2}\circ R_{l_1}$ has magnitude $2d$ in the direction perpendicular to the lines.

Step2: Calculate the distance between the lines

The line $k$ has the equation $x=-2$ and the line $t$ has the equation $x = - 1$. The distance $d$ between the two parallel lines $x=-2$ and $x=-1$ is $\vert-1-(-2)\vert=1$.

Step3: Determine the translation vector

Since we are finding $R_t\circ R_k$, the translation vector $\vec{v}$ has a magnitude of $2d$. Here, $d = 1$, so the magnitude of the translation vector is $2$. The direction of the translation is from the first - reflected - over line ($k$) to the second - reflected - over line ($t$). In the $x$ - direction (perpendicular to the vertical lines $x=-2$ and $x=-1$), the translation is in the positive $x$ - direction. So the translation $T$ can be written as a translation by the vector $\vec{v}=(2,0)$.

Answer:

The composition $R_t\circ R_k$ is a translation $T$ by the vector $(2,0)$