Professor Futrelle

Version of 17 January 2005

Due Friday, February 4th

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.

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].

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 = [4,0], B = [0,3] 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 [3,3] 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= -π/4.
5. Assume you have a triangle whose vertices are A=[10,0], B=[8,2] and C=[8,-2]. Draw it and then draw its appearance when rotated by -π/4 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= +π/4 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 [4,4], show that the product ProdA = Ra x Tc is not equal to to the product ProdB = Tc x Ra. Draw how the point p = [2,3] 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 P10=[10,10,10]. Apply the matrices in sequence one-by-one and then compute their product and apply it to P10 also.
12. Write out the translation matrix for [5,-7,3] and apply it to the 3d point P3=[4,7,6]. Is one of the transformed coordinates 0, as you'd expect?