Science:Math Exam Resources/Courses/MATH152/April 2022/Question A18/Solution 1

Let ${\textstyle L1}$ be the line that makes an angle of ${\textstyle 30^{\circ }}$ with the positive ${\textstyle x_{1}}$-axis, and ${\textstyle L2}$ be the line that makes an angle of ${\textstyle 50^{\circ }}$ with the positive ${\textstyle x_{1}}$-axis. We sketch the sequence of reflections below, keeping the original orientation of the axes in the background, labelled as $x_{1}(1)$, $x_{2}(1)$. In the sketch, the image of the $x_{1}(1)$-, $x_{2}(1)$-axes after ${\mathit {Ref}}_{30^{\circ }}$ is labelled as $x_{1}(2)$, $x_{2}(2)$. The reflection ${\mathit {Ref}}_{50^{\circ }}$ is then applied to the $x_{1}(2)$-, $x_{2}(2)$-axes, resulting in the $x_{1}(3)$-, $x_{2}(3)$-axes.

Math 152 2022 Q18 solution figure

After the reflection about ${\textstyle L2}$, we now look at the angle between the axes ${\textstyle x_{1}(1)}$ and ${\textstyle x_{1}(3)}$, this is ${\textstyle \theta =50^{\circ }-10^{\circ }=40^{\circ }}$ (shown in purple). The ${\textstyle x_{1}}$- and ${\textstyle x_{2}}$-axes are now ${\textstyle \theta =40^{\circ }}$ off from their original orientation (taking counterclockwise rotation to be positive).