Let H denote the plane in R3 that is given by the equation 3

Let H denote the plane in R^3 that is given by the equation 3x - 2y + z = 0. Let x denote the point with coordinates (2,1,-1).

c) Find the standard matrices for the orthogonal projection onto H and the reflection across H. State the matrices in the form A = (1/M) B where M is an integer and B is a matrix with integer entries.

Need projection AND reflection in form specified above WITH STEPS I need to understand how to do this.

Solution

We need to first find the orthogonal projection onto the plane H given by the equation 3x2y+z=0,

it is equal to the identity matrix minus the orthogonal projection onto H, which is easier to compute.

Now H is the span of the normal vector v=(3,2,1) (asp er the equation of the plane)

, and the orthogonal projection onto which is given by

x(vx)(vv)v, and whose matrix is

One normal vector to the plane is n=(3,2,1).

We want to take a point (x,y,z)R3,

so consider the line through this point with direction n, and see where it hits the plane. We have the line:

X(t)=(x+3t,y2t,z+t),tR.

We want t0 such that X(t0) satisfies the plane equation. So the relation we have is:

3(x+3t0)2(y2t0)+(z+t0)=03x2y+z+14t0=014t0=3x+2y-z.

or, t0=-(3x-2y+z)/14   ...i

substituting the value of t0 for any point X(t0) we will get

X(t0) =(5x+6y-3z)/14, (6x+10y+z)/14,(-3x+2y+13z)/14)

Thus for X(1,0,0) =(5/14,6/14,-3/14)

for X(0,1,0)=(6/14,10/14,2/14)

and X(0,0,1)=(-3/14,1/14,13/14)

so the orthogonal matrix will be given by

The inverse of the matrix will be the reflection of the point (2,1,-1) across H.