MAT188-WRITTEN-HOMEWORK 2, Oct 12th, 11:59 PM 2 Problem 1. Remember the Japanese puzzle from Homework 1. The puzzle is made from three identical pieces which are to be put together to form a 3 × 3 × 3 cube. The pieces are easy to design, but fitting them in a cube is surprisingly difficult. So you ask your friend for a hint. Your friend shares a few images from the solved puzzle with you that they took from different angles. You set up a coordinate system as shown in these images. 4 Z N FIGURE 0.1. Some views of the solved puzzle 2 Your friend has set up a fixed tripod with a camera attached, facing the back half of the xz-plane (y = 0). The rest of the images they send you are all taken from this fixed position. In the first shot you see the front of the letter C (using HW1 terminology) in red. Here the y-axis is behind the cube, perpendicular to the xz plane and hence you don't see it. MAT188-WRITTEN-HOMEWORK 2, Oct 12th, 11:59 PM 3 a Write the standard matrix of a linear transformation that transforms the first view (with the front view of the red letter C) to the second view (with the front view of the green letter C). Call it A. b Write the standard matrix of a linear transformation that transforms the second view (with the front view of the green letter C) to the third view (with the front view of the purple letter C). Call it B c Write the standard matrix of a linear transformation that transforms the first view (with the front view of the red letter C) directly to the third view (with the front view of the purple letter C). Call it D. What is the relationship between this matrix and the previous ones? You notice a symmetry in these three views of the solved puzzle. You remember the three pieces of the puzzle were identical in shape. That makes you think that the position of the pieces in the solved puzzle are interchangeable. That is: where the red piece lies now in Figure 0.1, could be lying a purple piece. So you decide to find a transformation that when applied to the purple piece in Figure 0.1 moves it to the position of the red piece in Figure 0.1. (1) Let p be the position vector of the point p marked in Figure 0.1. Find a vector 5 such that p+b=0. (2) Consider the transformation F(x) = +6, where b is the vector you found in the previous part. What is F(0)? Is F a linear transformation? Justify. (3) Let B denote the set of vectors whose tips when in standard position, create the purple piece in Figure 0.1. You don't need to describe B in set notation (even though you can if you wish). Sketch a picture (by hand or your choice of software) that shows {F(x) | x = B}, that is, the image of B under F. Mark the origin and the axis on your sketch. Mark and find the coordinates of all the corners of the front of the purple letter C. (4) Compare you image with this one. Is it the same? The other pieces are moved away to make room for the purple piece. NU The next image is taken from the same spot, after rotating the the cube 90 degrees clockwise about the z-axis. Notice that the origin is now blocked by the cube and not visible to you. And last image is taken after they rotate the cube again (from the second position) for 90 degrees counterclockwise about the x-axis. Once again, notice that the origin is behind the cube and you can't see it from this angle. FIGURE 0.2. The purple piece translated (5) Find two linear transformations T and S such that ToS moves the purple piece in Figure 0.2 to the position of the red piece in Figure 0.1. Describe T and S via their standard matrices. (6) Find a single linear transformation that moves the purple piece in Figure 0.2 to the position of the red piece in Figure 0.1. (7) Find a transformation that moves the purple piece in Figure 0.1 to the position of the red piece in Figure 0.1. Is this a linear transformation? Justify.

MAT188-WRITTEN-HOMEWORK 2, Oct 12th, 11:59 PM 2 Problem 1. Remember the Japanese puzzle from Homework 1. The puzzle is made from three identical pieces which are to be put together to form a 3 × 3 × 3 cube. The pieces are easy to design, but fitting them in a cube is surprisingly difficult. So you ask your friend for a hint. Your friend shares a few images from the solved puzzle with you that they took from different angles. You set up a coordinate system as shown in these images. 4 Z N FIGURE 0.1. Some views of the solved puzzle 2 Your friend has set up a fixed tripod with a camera attached, facing the back half of the xz-plane (y = 0). The rest of the images they send you are all taken from this fixed position. In the first shot you see the front of the letter C (using HW1 terminology) in red. Here the y-axis is behind the cube, perpendicular to the xz plane and hence you don't see it. MAT188-WRITTEN-HOMEWORK 2, Oct 12th, 11:59 PM 3 a Write the standard matrix of a linear transformation that transforms the first view (with the front view of the red letter C) to the second view (with the front view of the green letter C). Call it A. b Write the standard matrix of a linear transformation that transforms the second view (with the front view of the green letter C) to the third view (with the front view of the purple letter C). Call it B c Write the standard matrix of a linear transformation that transforms the first view (with the front view of the red letter C) directly to the third view (with the front view of the purple letter C). Call it D. What is the relationship between this matrix and the previous ones? You notice a symmetry in these three views of the solved puzzle. You remember the three pieces of the puzzle were identical in shape. That makes you think that the position of the pieces in the solved puzzle are interchangeable. That is: where the red piece lies now in Figure 0.1, could be lying a purple piece. So you decide to find a transformation that when applied to the purple piece in Figure 0.1 moves it to the position of the red piece in Figure 0.1. (1) Let p be the position vector of the point p marked in Figure 0.1. Find a vector 5 such that p+b=0. (2) Consider the transformation F(x) = +6, where b is the vector you found in the previous part. What is F(0)? Is F a linear transformation? Justify. (3) Let B denote the set of vectors whose tips when in standard position, create the purple piece in Figure 0.1. You don't need to describe B in set notation (even though you can if you wish). Sketch a picture (by hand or your choice of software) that shows {F(x) | x = B}, that is, the image of B under F. Mark the origin and the axis on your sketch. Mark and find the coordinates of all the corners of the front of the purple letter C. (4) Compare you image with this one. Is it the same? The other pieces are moved away to make room for the purple piece. NU The next image is taken from the same spot, after rotating the the cube 90 degrees clockwise about the z-axis. Notice that the origin is now blocked by the cube and not visible to you. And last image is taken after they rotate the cube again (from the second position) for 90 degrees counterclockwise about the x-axis. Once again, notice that the origin is behind the cube and you can't see it from this angle. FIGURE 0.2. The purple piece translated (5) Find two linear transformations T and S such that ToS moves the purple piece in Figure 0.2 to the position of the red piece in Figure 0.1. Describe T and S via their standard matrices. (6) Find a single linear transformation that moves the purple piece in Figure 0.2 to the position of the red piece in Figure 0.1. (7) Find a transformation that moves the purple piece in Figure 0.1 to the position of the red piece in Figure 0.1. Is this a linear transformation? Justify.

Delmar's Standard Textbook Of Electricity

7th Edition

ISBN:9781337900348

Author:Stephen L. Herman

Publisher:Stephen L. Herman

Chapter14: Basic Trigonometry And Vectors

Section: Chapter Questions

Problem 2PA: You desire to add a solar heating system to your swimming pool to extend the length of time you can...

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