Science:Math Exam Resources/Courses/MATH152/April 2017/Question A 16
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Question A 16 

For which value or values of θ in the interval does the matrix have as an eigenvector? Recall: is reflection through a line that makes an angle with the axis. 
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
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Hint 

Think about which points are fixed by a reflection through a line, and what does it mean for a vector to be an eigenvector. 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

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Solution 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Remeber that the only fixed points (eigenvectors) of a reflection are those which are on the line used for the reflection. Therefore is on the line whose angle we are looking for. Let us denote our line in the generic form . Since the line must pass through the origin, thus and since is on the line, satisfies . Therefore, the line we are looking for is given by the equation . We know that the slope of the line is the same as the tangent of the angle: therefore . Answer: . 
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