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

MATH152 April 2022

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[hide] Question A18

Let ${\textstyle {\mathit {Ref}}_{\alpha }}$ be the reflection of the plane ${\textstyle \mathbb {R} ^{2}}$ across the line ${\textstyle L}$ that makes an angle ${\textstyle \alpha }$ with the positive ${\textstyle x_{1}}$-axis. The composition of two such reflections $${\mathit {Ref}}_{50^{\circ }}\circ {\mathit {Ref}}_{30^{\circ }}$$ is the rotation by an angle ${\textstyle \theta }$. Find this ${\textstyle \theta }$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

[show] Hint 1
Draw a picture of what these reflections represent. What does the picture look like when you apply the second reflection to the result of the first reflection?

[show] Hint 2
Here is a hack that allows for a quick solution just in this case. The question statement already tells us that the composite ${\mathit {Ref}}_{50^{\circ }}\circ {\mathit {Ref}}_{30^{\circ }}$ is a rotation, so we just need to compute what it does to a single vector, for example a unit vector that is fixed by ${\mathit {Ref}}_{30^{\circ }}$.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

[show] 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).

[show] Solution 2

We will follow the second hint. Let $v\in \mathbb {R} ^{2}$ be a vector that makes an angle of $30^{\circ }$ with the $x$-axis. Then the first reflection fixes $v$, so the composite ${\mathit {Ref}}_{50^{\circ }}\circ {\mathit {Ref}}_{30^{\circ }}$ takes $v$ into ${\mathit {Ref}}_{50^{\circ }}(v)$, which is a vector that makes an angle of 40º with $v$, in the counterclock-wise direction. Therefore, ${\mathit {Ref}}_{50^{\circ }}\circ {\mathit {Ref}}_{30^{\circ }}$ is equal to rotation by 40º.