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

MATH221 April 2013

  •  Q1  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q2 (d)  •  Q3  •  Q4  •  Q5  •  Q6  •  Q7 (a)  •  Q7 (b)  •  Q7 (c)  •  Q8  •  Q9  •  Q10  •  Q11  •  Q12 (a)  •  Q12 (b)  •  Q12 (c)  •  

   Other MATH221 Exams

  •  December 2008  •  December 2009  •  December 2007  •  April 2010  •  April 2009  •  December 2011  •  April 2013  •  

Question 12 (a)
Consider the plane x - y + z = 0. a) Find the 3x3 matrix P1 which represents projection of $\mathbb {R} ^{3}$ 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].

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.

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. 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 ${\frac {x\cdot v}{v\cdot v}}v$, 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 $R^{3}$ 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. $v=\left[{\begin{array}{ccc}1&-1&1\\\end{array}}\right]$. This gives $v\cdot v=1+1+1=3$. We are now ready to apply it to a standard basis vector, $\left[{\begin{array}{ccc}1&0&0\\\end{array}}\right]$, which gets mapped by projection to the vecotr ${\frac {1}{3}}\left[{\begin{array}{ccc}1&-1&1\\\end{array}}\right]$. Similarly for the other basis vectors. Thus our matrix for the projection is: ${\frac {1}{3}}\left[{\begin{array}{cccc}1&-1&1\\-1&1&-1\\1&-1&1\\\end{array}}\right]$

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Orthogonal projection, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag