Science:Math Exam Resources/Courses/MATH152/April 2013/Question B 01 (b)
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Question B 01 (b) 

Let be the line . Find the matrix A which represents , the reflection across the line L. 
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. 
Hint 1 

To reflect a vector about a line that goes through the origin, let be a vector in the direction of the line. The matrix which represents is given by 
Hint 2 

For an alternative solution, we can recover the rotation matrix A if we know how it acts on two linearly independent vectors. Consider a vector on the given line, and a vector orthogonal to that line. 
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.

Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. To reflect a vector about a line that goes through the origin, let be a vector in the direction of the line. The matrix which represents is given by Since is on the line, we have 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. We can alternatively consider what a reflection does to vectors. Any vector pointing along the line 3x=4y, <4,3>, for example, will be reflected onto itself. Any vector perpendicular to <4,3> will point in the completely opposite direction after reflection. For example, <3,4> will become <3,4>. Note that these two vectors are linearly independent. Thus if A denotes the reflection, and , which we can write as a matrix equation
Thus 