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7. (10 points) Consider the cube with the eight vertices (+1, +1, +1). Let A, B,...
linear algebra: Consider the cube with the eight vertices (±1, ±1, ±1). Let A, B, and C be the midpoints of the three edges that join with the vertex (1, 1, 1). (a) What are the coordinates of A, B, and C? (b) What is the equation of the plane through A, B, and C? (c) What is the angle between the plane in (b) and the face x = 1 of the cube?
Consider the unit cube with vertices (corner points) (0, 0, 0),
(0, 1, 0), (1, 0, 0), (1, 1, 0), (0, 0, 1), (0, 1, 1), (1, 0, 1),
(1, 1, 1). Let S be the boundary of the cube minus (i.e. not
including) the bottom square (the side which lies in the xy plane).
Orient S with the normal which points out of the cube. Let F =
<− y , x , y^2e^x . Evaluate (curl F) ·...
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle between v and w. (b) Consider the projection proj, w of w onto v, and then project this projection on u to get proju (proj, w). Is this necessarily equal to the projection proj, w of w on u? Prove or give a counterexample. (c) Find the volume of the parallelepiped with edges formed by u-(2,5,c), v (1,1,1) and w...
Question 2: Consider the points located at A(1,1,1), B(2,2,3) and C(6,1,10). a) Find the true angle ABC with B at the vertex. b) Find the apparent angle A'B'C' when ABC is projected orthogonally onto the x-y plane. c) Find the apparent angle A"B"C" when ABC is projected orthogonally onto the X-z plane. B
Consider the following weighted, directed graph G. There are 7 vertices and 10 edges. The edge list E is as follows:The Bellman-Ford algorithm makes |V|-1 = 7-1 = 6 passes through the edge list E. Each pass relaxes the edges in the order they appear in the edge list. As with Dijkstra's algorithm, we record the current best known cost D[V] to reach each vertex V from the start vertex S. Initially D[A]=0 and D[V]=+oo for all the other vertices...
3. (2 Points) Let Q be the quadrilateral in the ry-plane with vertices (1, 0), (4,0), (0, 1), (0,4). Consider 1 dA I+y Deda (a) Evaluate the integral using the normal ry-coordinates. (b) Consider the change of coordinates r = u-uv and y= uv. What is the image of Q under this change of coordinates?bi (c) Calculate the integral using the change of coordinates from the previous part. Change of Variables When working integrals, it is wise to choose a...
Integrate A=(sinx, yx, cosz) over the outer surface of a cube with side length 1. For simplicity, place one vertex of the cube at the origin and align its three neighboring edges along the positive x, y, z axis, respectively. For each face of the cube, calculate A•n before doing the integration, where n is the norm of that face (the unit vector perpendicular to the surface that points outward).
Consider the graph at right. 17 15 [a] In what order are the vertices visited using DFS 319 1 starting from vertex a? Where a choice exists, use alphabetical order. What if you use BFS? [b A vertex x is "finished" when the recursive call DFS (x) terminates. In what order are the vertices finished? (This is different from the order in which they are visited, when DFS (x) is called.) [c] In what order are edges added to the...
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Let the deformation where ai, a2, and a3 are constants be applied to the unit cube of material shown in the sketch. Determine (a) the deformed length l of diagonal OC, (b) the angle between edges OA and OG after deformation, (c) the conditions which the constants must satisfy for the deformation to be possible (1) the material is incompressible, (2) the angle between elements OC and OB is to remain unchanged. x,,x, X2 , X2...
Q1. Given the points A: (0,0,2), B: (3,0,2), C: (1,2,1), and D: (2, 1,4 a) Find the cross product v - AB x AC. b) Find the equation of the plane P containing the triangle with vertices A, B, and C c) Find u the unit normal vector to P with direction v d) Find the component of AD over u and the angle between AD and u, then calculate the volume of the parallelepiped with edges AB, AC, AD...