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(1 point) In projecting an image onto the xy-plane, suppose that the viewing point (the center of...
A unit cube as shown in Figure Q1 is undergoing the transformations described in (i) and...
A unit cube as shown in Figure Q1 is undergoing the transformations described in (i) and (ii) respectively. Sketch the resultant object with coordinates of each vertex after each transformation. (a) Z (0,1,1) (1,1,1) (0,0,1) (1,0,1) (0,0,0) (1,1,0) (1,0,0) Figure Q1 Transformation (i) (6 marks) 1. A Uniform scale by a factor of 2 2. Followed by a rotation about the-axis in counter-clockwise direction by 90 degrees 3. Followed by a transformation moving in the direction of < 2, 1,...
3. The projection theorem we learned in class focuses on projecting onto subspaces. In general, W...
3. The projection theorem we learned in class focuses on projecting onto subspaces. In general, W, the set you are projecting onto, does not actually have to be subspace of V. (a) Let a and b be nonzero vectors in an inner product space (V,(,)) with a not a scalar multiple of b. Define W-(a+ bt where t E R} which is a subset of V. Show that W is not a subspace of V. (b) Given u in V,...
In the figure (Figure 1), the following EN and XY coordinates for points A through C...
In the figure (Figure 1), the following EN and XY coordinates for points A through C are given. Determine a 2-D conformal coordinate transformation, to convert the XY coordinates into the EN system. State Plane Coordinates (m) Arbitrary Coordinates (m) Point A 651779.323 290831.221 5804.32 3573.76 B 651369.152 290542.892 3543.59 2773.42 3857.64 3749.59 Figure 〈 1011 〉 Eg-EA アTA Part A What is the scale factor? Express your answer to six significant figures. View Available Hint(s) 8- Submit Part B...
linear algebra Remember we were able to express rotations and reflections, which are geometric transformations, using...
linear algebra
Remember we were able to express rotations and reflections, which are geometric transformations, using a linear transformation T, the coef- ficient matrix corresponding to the geometric transformation (r. y) (r', ) (a) What problem do you encounter with translations (r. y) (r+ h.y+k)? To handle this problem, We let the vector (x, y1 ) in R2 correspond to the vector (x1, y1, 1), and conversely. (In effect, we're projecting the :xy-plane onto the plane 1) introduce homogeneous coordinates....
10. (This topic is not covered on exam 3) moments about the axes and the center of mass. Mass, kg Location, m. (S,1) (-3.2) (1-1) a. A system of point masses (kg, meters) is distributed in the xy...
10. (This topic is not covered on exam 3) moments about the axes and the center of mass. Mass, kg Location, m. (S,1) (-3.2) (1-1) a. A system of point masses (kg, meters) is distributed in the xy-plane as follows. Find the (1,0) (4,-2) b. Find the centroid of the triangular region with vertices (0,0), (3,0), and (5,0). c. Find the center of mass of a thin homogeneous plate forming a sector of a circle of radius r and angle...
Suppose E is the half-cylinder described by x^2 + y^2 = 1 between z = 4 and the xy-plane where y ≥ 0. Suppose further that the density at each point in E is proportional to the distance from the z-axi...
Suppose E is the half-cylinder described by x^2 + y^2 = 1 between z = 4 and the xy-plane where y ≥ 0. Suppose further that the density at each point in E is proportional to the distance from the z-axis. (a) Find an expression for the mass of E as a triple integral. Then briefly explain why this integral is difficult to compute. (b) (8 points) Describe the solid E using cylindrical coordinates.Then express the mass of E as...
1. In the figure below, three point particles are fixed in place in the xy plane. The three partiles sit on the corners...
1. In the figure below, three point particles are fixed in place in the xy plane. The three partiles sit on the corners of an equilateral triangle with sides of length a = 2.50 mm. Particle 1 has a mass m1 = 12.0 kg, particle 2 has a mass m2 = 18.0 kg, and particle 3 has a mass m3 = 15.0 kg. m --- -- 1 m3 (a) What is the magnitude and direction of the net gravitational force...
2. (1 Point) Let r-2u and y-3u. (a) Let R be the rectangle in the uv-plane defined by the points (0,0), (2,0), (2,...
2. (1 Point) Let r-2u and y-3u. (a) Let R be the rectangle in the uv-plane defined by the points (0,0), (2,0), (2,1), (0 , 1). Find the area of the image of R in the ry plane? (b) Find the area of R by computing the Jacobian of the transformation from uv-space to xy-space Change of Variables When working integrals, it is wise to choose a coordinate system that fits the problem; e.g. polar coordinates are a good choice...
Consider 1-2 Vr? + y + 3 LLL da dydar. V1-38-98 V +y + y2 +22...
Consider 1-2 Vr? + y + 3 LLL da dydar. V1-38-98 V +y + y2 +22 +y +22-2 the origin to the point (2, y, ) makes with the z-axis is a new angle which we will label o, and we label the length of the line segment p. We can now determine the remaining side-lengths of our new triangle. Let us try to label our point (2, y, z) in only p and 6. Our labeled triangle gives us...
A unit cube as shown in Figure Q1 is undergoing the transformations described in (i) and (ii) respectively. Sketch the resultant object with coordinates of each vertex after each transformation. (a) Z (0,1,1) (1,1,1) (0,0,1) (1,0,1) (0,0,0) (1,1,0) (1,0,0) Figure Q1 Transformation (i) (6 marks) 1. A Uniform scale by a factor of 2 2. Followed by a rotation about the-axis in counter-clockwise direction by 90 degrees 3. Followed by a transformation moving in the direction of < 2, 1,...
3. The projection theorem we learned in class focuses on projecting onto subspaces. In general, W, the set you are projecting onto, does not actually have to be subspace of V. (a) Let a and b be nonzero vectors in an inner product space (V,(,)) with a not a scalar multiple of b. Define W-(a+ bt where t E R} which is a subset of V. Show that W is not a subspace of V. (b) Given u in V,...
In the figure (Figure 1), the following EN and XY coordinates for points A through C are given. Determine a 2-D conformal coordinate transformation, to convert the XY coordinates into the EN system. State Plane Coordinates (m) Arbitrary Coordinates (m) Point A 651779.323 290831.221 5804.32 3573.76 B 651369.152 290542.892 3543.59 2773.42 3857.64 3749.59 Figure 〈 1011 〉 Eg-EA アTA Part A What is the scale factor? Express your answer to six significant figures. View Available Hint(s) 8- Submit Part B...
linear algebra
Remember we were able to express rotations and reflections, which are geometric transformations, using a linear transformation T, the coef- ficient matrix corresponding to the geometric transformation (r. y) (r', ) (a) What problem do you encounter with translations (r. y) (r+ h.y+k)? To handle this problem, We let the vector (x, y1 ) in R2 correspond to the vector (x1, y1, 1), and conversely. (In effect, we're projecting the :xy-plane onto the plane 1) introduce homogeneous coordinates....
10. (This topic is not covered on exam 3) moments about the axes and the center of mass. Mass, kg Location, m. (S,1) (-3.2) (1-1) a. A system of point masses (kg, meters) is distributed in the xy-plane as follows. Find the (1,0) (4,-2) b. Find the centroid of the triangular region with vertices (0,0), (3,0), and (5,0). c. Find the center of mass of a thin homogeneous plate forming a sector of a circle of radius r and angle...
1. In the figure below, three point particles are fixed in place in the xy plane. The three partiles sit on the corners of an equilateral triangle with sides of length a = 2.50 mm. Particle 1 has a mass m1 = 12.0 kg, particle 2 has a mass m2 = 18.0 kg, and particle 3 has a mass m3 = 15.0 kg. m --- -- 1 m3 (a) What is the magnitude and direction of the net gravitational force...
2. (1 Point) Let r-2u and y-3u. (a) Let R be the rectangle in the uv-plane defined by the points (0,0), (2,0), (2,1), (0 , 1). Find the area of the image of R in the ry plane? (b) Find the area of R by computing the Jacobian of the transformation from uv-space to xy-space Change of Variables When working integrals, it is wise to choose a coordinate system that fits the problem; e.g. polar coordinates are a good choice...
Consider 1-2 Vr? + y + 3 LLL da dydar. V1-38-98 V +y + y2 +22 +y +22-2 the origin to the point (2, y, ) makes with the z-axis is a new angle which we will label o, and we label the length of the line segment p. We can now determine the remaining side-lengths of our new triangle. Let us try to label our point (2, y, z) in only p and 6. Our labeled triangle gives us...
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