## Professor Futrelle

Version of 10 September 2005

### Due in class, Monday, September 26th (Quiz on the 29th)

Do the following problems by hand, on paper. No calculator is needed or should be used. For example, a value such as √2 should be kept in that form and certainly not written down as 1.41421356237309504880, etc. In drawing items based on such values, a reasonable approximation suffices. The first three questions can be done based on material in Chapter 2. The remainder are based on the material in Chapter 5, and especially, the material in Chapter 6.

For convenience in writing out the problems below, vectors are written in row format, in contrast to the book's column format, as [x,y] or [x,y,z] or for homogeneous coordinates [x,y,1] or [x,y,z,1]. You should write your answers (and exam answers) using vectors in column format.

When you are asked to do a transformation step-by-step, write, draw and, comment on the results of each step.

1. Draw the vectors A = [8,0], B = [0,6] and the sum C of A and B. Using the coordinates of C, compute its length.
2. As in Problem 1, but draw and compute the difference, C - B.
3. Write out the normalized form of the vector [7,2] and show how you computed it.
4. Write out the general form of the rotation matrix R, as well as its value Rb for θ = -3/4π.
5. Assume you have a triangle whose vertices are A=[6,0], B=[4,3] and C=[4,-3]. Draw it and then draw its appearance when rotated by -3/4π around the origin.
6. Show that applying Rb in Problem 4 to the vertices of the triangle in Problem 5 gives the same result as what you drew in 5.
7. Write the value of the rotation matrix Ra, for θ = +3/4π and show that Ra x Rb is the identity by computing each element. Show that Rb x Ra gives the same result.
8. Write out the 3x3 translation matrix Ta for a translation by [tx,ty]. By multiplying out the components, show in general that the product of Ta and Tb (for a translation [-tx,-ty]) is the identity matrix.
9. For a translation Tc, by [5,5], show that the product ProdA = Ra x Tc is not equal to to the product ProdB = Tc x Ra. Draw how the point p = [3,2] should be transformed by the two transforms matrices, ProdA and ProdB, doing them step-by-step, using Ra and Tc separately. Then transform p by actually multiplying it properly by the two different composite matrices, ProdA and ProdB. Your results should agree with what you just drew.
10. Consider the two points [5,6] and[5,4]. Use a single matrix that is the product (in order applied) of a translation by [-5,-5], a rotation by +π/2, and a translation by [5,5] and apply it two both points. What did you expect to happen and did you get the result you expected?
11. Write out the 3D transforms for +π/2 rotations around the z axis (transform Tz) and around the x axis (transform Tx). Draw and compute the result of applying Tz and then Tx to the point P5=[5,5,5]. Apply the matrices in sequence one-by-one and then compute their product and apply it to P5 also.
12. Write out the translation matrix for [5,-4,3] and apply it to the 3d point P3=[4,6,-3]. Is one of the transformed coordinates 0, as you'd expect?