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Problem 5 Find the rotation matrix that will line up the orthogonal vectors, and = - ( ) along the y-and 2-axes, respectively To rotate A into the y-axis you have to apply successive rotations to zero out the z and 2 components. Follow these steps: (a) Draw a picture that shows the vector projected into the zy-plane. (This projected vector has r and y components that are both I unit of length.) (b) To rotate the projected vector about the 2-axis, mark the angle between the y-axis and the projected vector A. Using the picture, determine the sine and cosine of the angle you marked. The rotation matrices use the sine and cosine of this angle so you may not actually need to know the value of the angle. (Read questions carefully!) (c) Generate the rotation matrix (call it Ri ) for a rotation about the z-axis and apply it to the vector A: Determine A' - R A. If you did this correctly, the I-component will be 0 at the end. Take care to note this is a counterclockwise (positive ) rotation. (d) The final steps now require a rotation about the r-axis to zero out the y-component As above, draw a picture with the new y and 2 components. Mark the angle be- tween the rotated vector and the y-axis. Use the picture to determine the sine and cosine of the angle. Write the rotation matrix (for a rotation about the r-axis; call it R2). Apply the rotation matrix to the rotated vector. A" - RZA'. If you did this correctly, you will get A" -V3y. Note that this step requires a clockwise (negative) rotation. (e) The total rotation matrix for this process is R - R R To proceed, since A and B are orthogonal you can apply BY - RB. When you do this you'll see that it does not zero out the I and y components. You'll need one more rotation to align the vector B along the 2-axis Hint: you will have to do a rotation about the y-axis to align the vector B with the z-axis; call this R,; pay close attention when you do this step). Notice that the total rotation matrix is now R - R3R2R. Check that this still rotates A into the y-axis