problem 5:

Problem 5. Let H be the plane x + y + z = 0, let n be a unit normal vector to H, and let L be the line through the origin which is perpendicular to H. Define two 3 x 3 matrices, as follows: P = nnT [7] (in -1, and Q =13 - a. If a is an arbitrary 3D vector, show that Pa is parallel to L, Qa is parallel to H, and & = P& + Qx. b. Let F be the matrix which reflects vectors orthogonally across H. Draw the vectors Pr, Qr, and Fx in the following diagram: L H c. Give constants a and b such that the matrix F = ap +bQ reflects vectors across H. d. Compute the matrix F, and verify that it is an orthogonal matrix. e. Find the matrix of F with respect to the basis B from problem 4. (Hint: no computation is required, just draw the basis vectors and their transforms in your diagram.) Note: You may find it enlightening to verify that part a works for any plane through the origin, not just H. Therefore, this method can be used to compute any 3-dimensional reflection matrix