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.

- 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.
- As in Problem 1, but draw and compute the difference, C - B.
- Write out the normalized form of the vector [7,2] and show how you computed it.
- Write out the general form of the rotation matrix R, as well
as its value R
_{b}for θ = -3/4π. - 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.
- Show that applying R
_{b}in Problem 4 to the vertices of the triangle in Problem 5 gives the same result as what you drew in 5. - Write the value of the rotation matrix R
_{a}, for θ = +3/4π and show that R_{a}x R_{b}is the identity by computing each element. Show that R_{b}x R_{a}gives the same result. - Write out the 3x3 translation matrix T
_{a}for a translation by [tx,ty]. By multiplying out the components, show in general that the product of T_{a}and T_{b}(for a translation [-tx,-ty]) is the identity matrix. - For a translation T
_{c}, by [5,5], show that the product Prod_{A}= R_{a}x T_{c}is not equal to to the product Prod_{B}= T_{c}x R_{a}. Draw how the point p = [3,2] should be transformed by the two transforms matrices, Prod_{A}and Prod_{B}, doing them step-by-step, using R_{a}and T_{c}separately. Then transform p by actually multiplying it properly by the two different composite matrices, Prod_{A}and Prod_{B}. Your results should agree with what you just drew. - 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?
- Write out the 3D transforms for +π/2 rotations around the z axis
(transform T
_{z}) and around the x axis (transform T_{x}). Draw and compute the result of applying T_{z}and then T_{x}to the point P_{5}=[5,5,5]. Apply the matrices in sequence one-by-one and then compute their product and apply it to P_{5}also. - Write out the translation matrix for [5,-4,3] and apply it to the 3d
point P
_{3}=[4,6,-3]. Is one of the transformed coordinates 0, as you'd expect?

Go to CSU540 home page. or RPF's Teaching Gateway or homepage