3.6 Exercises

1. Prove that the rows of R_y are orthonormal.

2. Prove that .

3. Compute:

    

   

   Does the translation translate points? Does the translation translate vectors? Why does it not make sense to translate the coordinates of a vector in standard position?

4. Verify that the given scaling matrix inverse is indeed the inverse of the scaling matrix; that is, show, by directly doing the matrix multiplication, SS⁻¹ = S⁻¹S = I. Similarly, verify that the given translation matrix inverse is indeed the inverse of the translation matrix; that is, show that TT⁻¹ = T⁻¹T = I.

5. Suppose that we have frames A and B. Let p_A = (1, −2, 0) and q_A = (1, 2, 0) represent a point and force, respectively, relative to frame A. Moreover, let Q_B = (−6, 2, 0),  and w_B = (0, 0, 1) describe frame A with coordinates relative to frame B. Build the change of coordinate matrix that maps frame A coordinates into frame B coordinates, and find p_B = (x, y, z) and q_B = (x, y, z). Draw a picture on graph paper to verify that your answer is reasonable.

6. Redo , but this time scale the square 1.5 units on the x-axis, 0.75 units on the y-axis, and leave the z-axis unchanged. Graph the geometry before and after the transformation to confirm your work.

7. Redo , but this time rotate the square −45º clockwise about the y-axis (i.e., 45º counterclockwise). Graph the geometry before and after the transformation to confirm your work.

8. Redo , but this time translate the square −5 units on the x-axis, −3 units on the y-axis, and 4 units on the z-axis. Graph the geometry before and after the transformation to confirm your work