Science:Math Exam Resources/Courses/MATH221/April 2013/Question 12 (a)/Solution 1

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: ${\frac {1}{3}}\left[{\begin{array}{cccc}1&-1&1\\-1&1&-1\\1&-1&1\\\end{array}}\right]$