MATLAB (SOLVE IT ON MATLAB SO I CAN COPY THE CODE)

Let be the plane described by the equation 2x - y + z = 0. In this problem, you will produce a matrix P such that Px is the orthogonal projection of the vector x on the plane . Note that the plane belongs to the three-dimensional space, so the matrix P must be of type 3 3.

(a) Construct a normal vector to the plane and call it n. In this task, it is important that we represent n as a column vector. Remind yourself what the projection formula for orthogonal projection of a vector x on n looks like. Now construct the matrix N according to N =( (1)/ (n)2) * nnT It is now claimed that the matrix image TN with the matrix N performs orthogonal projection on the vector n. Investigate this by calculating Nx for a few different vectors x and comparing with the project (x). Use MATLAB for your calculations.

(b) Define the orthogonal projection of the vector x on the plane to be the componentvektorn x:s of the vector x perpendicular to the vector n. Write down a formula for this projection. Now construct the matrix P such that Px is the orthogonal projection of x on the plane . Use the matrix N to construct P. Check if the result is correct by calculating Px and comparing with x for some different vectors x.

(c) In this sub-task, you will work with paper and pen to show that the matrices N and P really calculate what is claimed in the task. Thus, show that Nx is the projn (x) for each vector x in space, and the equivalent for the matrix P. It may be helpful to remember that the scalar product can also be calculated by multiplying matrices: u v = uT v , where u and v are column vectors and the right-hand side is matrix multiplication.