2 In C, (x, y) = xAy* is an inner product, where A = i ). Compute (x, y) -i 3 for r = (1+ i, 3i) and y = (-3+ i,3+ 2i).
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- Use the inner product u,v=2u1v1+u2v2 in R2 and Gram-Schmidt orthonormalization process to transform {(2,1),(2,10)} into an orthonormal basis.Evaluate the triple integral ∭ExydV where E is the solid tetrahedon with vertices (0,0,0),(2,0,0),(0,6,0),(0,0,4)Find the area of the portion of the paraboloid x = 4 - y2 - z2 that lies above the ring 1<=y2 + z2<=4 in the yz-plane.
- Evaluate the triple integral. 3xyz dV, where T is the solid tetrahedron with vertices (0, 0, 0), (1, 0, 0), (1, 1, 0), and (1, 0, 1)Find the triple integral: 24xy^2z^3dV over the box B = [0, a] × [0, b] × [0, c]In isosceles triangle XYZ, XY = YZ = 10, and XZ = 16. Where C is the centroid of triangle XYZ, find the distance from C to side XZ of the triangle.
- Find the orthogonal trajectories of the conicoid (x + y)z = 1 of the conics in which it is cut by the system of the plane x − y + z = K where K is a parametersThe upper half of the unit sphere x2+y2+z2=1 is z=Square root of 1-x2-y2. Find its centroid?C is the hexagon in the xy-plane with vertices (5, −5), (5, 5), (0, 10), (−5, 5), (−5, −5), and (0, −10), directed counter-clockwise. Evaluate integralC (3y/(x^2+y^2) dx − 3x/(x^ 2 + y^2) dy)
- Let C be the positively oriented square with vertices (0,0), (3,0), (3,3), (0,3). Use Green's Theorem to evaluate the line integral ∫c 7y^2x dx+6x^2y dy.1. Solve by Cramer’s rule 3x + y + 4z = 11 4x – 4y + 6z = 11 6x – 6y = 3 2. Find the volume of tetrahedron given the following (1, 0, 1), (0, 1, 0), (0, 0, 1), and (1, 1, 1).I don't understand why the double integral of z dA over region D turns into [(64-4r^2)^(1/2)-(-(64-4r^2)^(1/2)]rdrdtheta, instead of just (64-4r^2)^(1/2) rdrdtheta since R {(r,theta) | 0 <= r <= 2; 0 <= theta <+ 2pi}.