Find the point on the plane 2 + 3y + 4z = 0 which is closest to the point (1,0, 4) in twodifferent ways:(a) Using the method of Lagrange multipliers, and(b) By finding an orthonormal basis for the plane and using that to obtain a projection matrix.

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Answered: Find the point on the plane 2x + 3y +…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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6. Project the vector b = (1, 4, 2) onto the plane S that contains the vectors a₁ = (1, 2, 1) and a2 = (1, 1,0). (i) Find the 3 x 3 projection matrix P onto S. (ii) Show that P2 = P. (iii) Find a vector whose projection onto S is the zero vector. (iv) Show that I - P projects onto the line perpendicular to S.
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#10: Let W be the plane with the equation 3x-y + 62 = 0. Find the standard matrix P for the orthogonal projection onto W. Use the following formula P = A(ATA)-¹AT, where the matrix A is constructed using any basis for W as its column vectors. Enter the first row of the matrix P (in order) into the answer box below, separated with commas. i.e., enter the values P11, P12, P13.
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Find the point on the plane 2x + 3y + 4z = 0 which is closest to the point (1,0, 4) in two %3D different ways: (a) Using the method of Lagrange multipliers, and (b) By finding an orthonormal basis for the plane and using that to obtain a projection matrix.