Use homogeneous coordinates for all of the following problems.

1) Construct the matrix that scales by (S1,S2,S3) relative to the point (C1,C2,C3).

2) Construct the matrix that magnifies the triangle (a,b,c) by the factor 3 while keeping vertex a fixed for a = (1,1,0), b = (2,2,0), and c = (4,-1,0).

3) Let R(a) denote the matrix that rotates (about the z axis) by angle a. Find an expression for R(a+b) and show that rotations commute; i.e., R(a)R(b) = R(b)R(a) = R(a+b).

4) Let A and B denote order-4 matrices, and let p be a point in homogeneous coordinates (4 by 1 matrix with last component equal to 1). Specify the number of multiplies required to compute (a) (A B)p and (b) A(B p). Note that no multiplies are required to compute the last row of (A B) or the last component of (B p). Explain why it is usually advantageous to combine transformations before applying them (despite the fact that case (a) requires more multiplications).

5) Show that translations and scalings commute among themselves but not with each other; i.e., T T' = T' T and S S' = S' S but T S and S T are generally not equal.