Science:Math Exam Resources/Courses/MATH221/April 2013/Question 12 (a)
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Question 12 (a) 

Consider the plane x  y + z = 0. a) Find the 3x3 matrix P_{1} which represents projection of onto a vector orthogonal to this plane. 
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 

What is the formula that projects one vector onto another? 
Hint 2 

To find the projection matrix it suffices to calculate the projections the canonical basis vectors [1,0,0], [0,1,0], [0,0,1]. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We recall that the formula to compute a projection onto a line spanned by the vector v is to take a vector x and mapping it to , that is you take the inner product of the vector x with the vector v divide it by the length of the vector v and apply this scalar to v. To get a matrix out of this, we can simply apply the projection to the standard basis of and see where each vector is mapped to. First we note that, for us, the vector v is the normal vector of the plane, i.e. . This gives . We are now ready to apply it to a standard basis vector, , which gets mapped by projection to the vecotr . Similarly for the other basis vectors. Thus our matrix for the projection is: 