Science:Math Exam Resources/Courses/MATH152/April 2010/Question A 19
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Question A 19 

If is the 2D counterclockwise rotation by and is the 2D reflection through the line , is ? Justify briefly. 
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 

Do both operations , have the same effect on all vectors in ? 
Hint 2 

The statement is not true. Try to find a counter example. 
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 

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Please rate my easiness! It's quick and helps everyone guide their studies. Consider a vector, v, that lies on the line . Applying to this vector will first rotate the vector so that it lies parallel to the axis, then reflect the vector over so that it lies parallel to the axis. Applying to this vector will first reflect the vector over the line , leaving it unchanged, then will rotate the vector so that it lies parallel to the axis. By this example, and have different effects on v and thus do not commute. i.e:
