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(a) Consider two lines on a plane in the homogeneous coordinate system l1 = (8,3,4)T and...
Math 32-_ Multivariable Calculus HW 3 (1) Consider the two straight lines L1 : (2-t, 3...
Math 32-_ Multivariable Calculus HW 3 (1) Consider the two straight lines L1 : (2-t, 3 + 2t,-t) and L2 : <t,-2 + t, 7-20 a) Verify that L1 and L2 intersect, and find their point of intersection. (b) Find the equation of the plane containing L1 and L2 (2) Consider the set of all points (a, y, z) satisfying the equation 2-y2+220. Find their intersection 0 and 2-0. Use that information to sketch a with the planes y =-3,-2,-1,0,...
3. (14 points) Given the lines: 21:2(t) = -3t – 1, y(t) = 2t +4, z(t)...
3. (14 points) Given the lines: 21:2(t) = -3t – 1, y(t) = 2t +4, z(t) =t+4 12: x(u) = 5 - 3u, y(u) = u +1, (u) = u +2 1. Determine whether li and ly are parallel, skew or intersect. If the lines intersect, find the point of intersection of li and 12. 2. If the lines intersect or are parallel, give an equation for the plane which contains both lines. If the lines are skew, find a...
(1 point) Using homogeneous coordinates, the matrix A which reflects the point (5,-4,-1) across t...
(1 point) Using homogeneous coordinates, the matrix A which reflects the point (5,-4,-1) across the x- axis, shifts the result by 1 in the x-direction, -1 in the y-direction, and -1 in the z-direction, and then expands the y coordinate by a factor of 5 is given by A= The image of (5,-4,-1) under this transformation is , and Z =
(1 point) Using homogeneous coordinates, the matrix A which reflects the point (5,-4,-1) across the x- axis, shifts the...
(1 point) In projecting an image onto the xy-plane, suppose that the viewing point (the center of...
(1 point) In projecting an image onto the xy-plane, suppose that the viewing point (the center of projection) is located at the point (0, 0, 11). Using homogeneous coordinates, a perspective projection matri A which projects the onto the xy-plane is given by Consider a triangle with comers at A (5,4,4), B (1,6,3), and C (3,3,7). Projecting this onto the xy-plane under this transformation gives a triangle with corners at the points A', B', and C' correspondingly where C' (Express...
34. Consider a two-dimensional Cartesian coordinate system and a two-dimensional uv-system with t...
34. Consider a two-dimensional Cartesian coordinate system and a two-dimensional uv-system with the coordinates related by y = (1/2)(112-U2). In general, Laplace's equation in two dimensions can be written as with ох ду Zi (a) In the xy-plane, sketch lines of constant u and constant v. (b) Express Laplace's equation using the uw-coordinates. (c) Use the method of separation of variables to separate Laplace's equation in the v-system and obtain the general solution for ų'(u, u).
34. Consider a two-dimensional...
Question 4 Consider the lines L, D=1+2, y = 2 – 3t, z = 2+t and...
Question 4 Consider the lines L, D=1+2, y = 2 – 3t, z = 2+t and L2 X = 3 - 4s, y=1+ 48, z = -3 + 48. We will use these lines for the questions 4 and 5. Are these lines parallel? Explain your answer below. B IV A - A - Ix E - C o o x G You HTML 11 x Ⓡ 5E T To 12pt Pan Question 5 9 pts Determine where these lines...
1. Use the following tited coordinate system to deduce the plane-stress t transformation equations: (40 pts)...
1. Use the following tited coordinate system to deduce the plane-stress t transformation equations: (40 pts) 2 dy d: dx Use the transformation equations to find and draw the principal stresses and maximum shear stresses and directions. (40 pts) 2. 50 MPa 30 MPa 100 MPa
2 Homogeneous coordinates Recall that an affine function is of the form f^x) Mx + t for a matrix M and vector t. Homoge...
2 Homogeneous coordinates Recall that an affine function is of the form f^x) Mx + t for a matrix M and vector t. Homogeneous coordinates are frequently used to represent affine functions in robotics and 3D graphics. We define the function H by and if f-x) Mxtt where then C0 a. Some vectors are valid homogeneous representations of vectors, and some are not. Explain how to tell if some vector y-0 is the homogeneous representation of some other vector -y...
Consider the homogeneous system 0 ' = Až = a) T, a ER -4 a (a)...
Consider the homogeneous system 0 ' = Až = a) T, a ER -4 a (a) Determine the eigenvalues of A in terms of a. (if any) where the qualitative nature of phase portrait for (b) Identify the values of a system changes. (c) For each value of a that you listed in part (b), sketch the phase portrait for a value of a that is slightly less than the value(s) that you identified. For example, if you identified only...
Consider the linear transformation T: Rn → Rn whose matrix A relative to the standard basis...
Consider the linear transformation T: Rn → Rn whose matrix A relative to the standard basis is given. A 2 2 (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) (A1, A2) -1 5 (b) Find a basis for each of the corresponding eigenspaces (c) Find the matrix A' for Trelative to the basis B', where B' is made up of the basis vectors found in part (b)
Math 32-_ Multivariable Calculus HW 3 (1) Consider the two straight lines L1 : (2-t, 3 + 2t,-t) and L2 : <t,-2 + t, 7-20 a) Verify that L1 and L2 intersect, and find their point of intersection. (b) Find the equation of the plane containing L1 and L2 (2) Consider the set of all points (a, y, z) satisfying the equation 2-y2+220. Find their intersection 0 and 2-0. Use that information to sketch a with the planes y =-3,-2,-1,0,...
3. (14 points) Given the lines: 21:2(t) = -3t – 1, y(t) = 2t +4, z(t) =t+4 12: x(u) = 5 - 3u, y(u) = u +1, (u) = u +2 1. Determine whether li and ly are parallel, skew or intersect. If the lines intersect, find the point of intersection of li and 12. 2. If the lines intersect or are parallel, give an equation for the plane which contains both lines. If the lines are skew, find a...
(1 point) Using homogeneous coordinates, the matrix A which reflects the point (5,-4,-1) across the x- axis, shifts the result by 1 in the x-direction, -1 in the y-direction, and -1 in the z-direction, and then expands the y coordinate by a factor of 5 is given by A= The image of (5,-4,-1) under this transformation is , and Z =
(1 point) Using homogeneous coordinates, the matrix A which reflects the point (5,-4,-1) across the x- axis, shifts the...
(1 point) In projecting an image onto the xy-plane, suppose that the viewing point (the center of projection) is located at the point (0, 0, 11). Using homogeneous coordinates, a perspective projection matri A which projects the onto the xy-plane is given by Consider a triangle with comers at A (5,4,4), B (1,6,3), and C (3,3,7). Projecting this onto the xy-plane under this transformation gives a triangle with corners at the points A', B', and C' correspondingly where C' (Express...
34. Consider a two-dimensional Cartesian coordinate system and a two-dimensional uv-system with the coordinates related by y = (1/2)(112-U2). In general, Laplace's equation in two dimensions can be written as with ох ду Zi (a) In the xy-plane, sketch lines of constant u and constant v. (b) Express Laplace's equation using the uw-coordinates. (c) Use the method of separation of variables to separate Laplace's equation in the v-system and obtain the general solution for ų'(u, u).
34. Consider a two-dimensional...
Question 4 Consider the lines L, D=1+2, y = 2 – 3t, z = 2+t and L2 X = 3 - 4s, y=1+ 48, z = -3 + 48. We will use these lines for the questions 4 and 5. Are these lines parallel? Explain your answer below. B IV A - A - Ix E - C o o x G You HTML 11 x Ⓡ 5E T To 12pt Pan Question 5 9 pts Determine where these lines...
1. Use the following tited coordinate system to deduce the plane-stress t transformation equations: (40 pts) 2 dy d: dx Use the transformation equations to find and draw the principal stresses and maximum shear stresses and directions. (40 pts) 2. 50 MPa 30 MPa 100 MPa
2 Homogeneous coordinates Recall that an affine function is of the form f^x) Mx + t for a matrix M and vector t. Homogeneous coordinates are frequently used to represent affine functions in robotics and 3D graphics. We define the function H by and if f-x) Mxtt where then C0 a. Some vectors are valid homogeneous representations of vectors, and some are not. Explain how to tell if some vector y-0 is the homogeneous representation of some other vector -y...
Consider the homogeneous system 0 ' = Až = a) T, a ER -4 a (a) Determine the eigenvalues of A in terms of a. (if any) where the qualitative nature of phase portrait for (b) Identify the values of a system changes. (c) For each value of a that you listed in part (b), sketch the phase portrait for a value of a that is slightly less than the value(s) that you identified. For example, if you identified only...
Consider the linear transformation T: Rn → Rn whose matrix A relative to the standard basis is given. A 2 2 (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) (A1, A2) -1 5 (b) Find a basis for each of the corresponding eigenspaces (c) Find the matrix A' for Trelative to the basis B', where B' is made up of the basis vectors found in part (b)
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