onLet √ € R³ and W = span ((1, 1, 0)², (1, −1,0)²'). With respect to the standard inner product, which of the following is the orthogonal projection of (3, 4, 5)² € R³ 0W?(1,-1,0)T(1,-1) ¹2(3, 4) T(3,4,0)T

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Answered: Let u € R³ and W = span ((1, 1, 0)",…

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.
A vector orthogonal to the plane through the points (-1,0,0), (0,0, -2), (2, 3, 4) is < 10, – 10,0 : < 6, – 10, 3 : < -2, – 10,9 : | < 2, – 10, 6 : -
Q3) Find the vector ž, which is perpendicular to vectors å = (2,3, –1) and 5 = (1,–2,3) and satisfies the relation ž. (2ï – j + R) = -6. А) (2, -6, —3) B) (-1,2,0) C) (-3,3,3) D) (2, –2, – 1) E) (1,–1,–1)
Express the given parametric equations of a line using bracket notation and also using i, j, k notation. (a) x = 6t, y = - 4 + 3t O (x, y) = (0, 4) +1 (6,3) ai + bj = -4j + t (6i + 3j) O (x, y) = (0, 4) + t (6, 3) ai + bj = 4j+ t (6i + 3j) O (x, y) = (0, 6) + (-4, 3) ai + bj = − 4j + t (6i + 3j) O(x, y) =(4,0) + t (3,6) ai + bj = −4i + t (3i + 6j) O (x, y) = (6,3)+1(0, - 4) ai + bj (6i + 3j) +t(-4j) (b)x= 4 + 6t,y=4+9t, z = 87t O (x, y, z) = (-4, 4, − 8) +1(-6, ai + bj + ck = 9,7) ( − 4i + 4j - 8 k) + t ( − 6i - 9j + 7k) O (x, y, z) = (4, ai + bj + ck = 4, 8) + t (6, 9, - 7) (4i − 4j + 8 k) + t (6i + 9j – 7k) O (x, y, z) = (6,9, −7) +1(4, - 4,8) ai + bj + ck = (6i + 9j − 7k) + t(4i - 4j + 8k) O (x, y, z) = (-6, 9, 7) + 1 - 4, 4, 8) ai+bj + ck = ( – 6i – 9j + 7k) + t( − 4i + 4j − 8k) O (x, y, z) = (4, 4, 8) + t (6,9,7) ai + bj + ck = (4i +4j + 8 k) + t (6i + 9j + 7k)
11. Assume u=(1,0,0,-1), vị=(3,2,7,3), v2=(2,1,3,2), and v3=(5,2,9,5). is in the orthogonal complement of W=span(v1,V2,V3). Show that u
given u = <14,5> and v = <1,4>, find the projection of u onto v (a) <31, 13> (b) <28,8> (c) <5,11> (d) <2,8> (e) <28,10>
Express the given parametric equations of a line using bracket notation and also using i, j, k notation. (a) x = 6t, y = - 4 + 3t O (x, y) = (0, - 4) + t (6,3) ai + bj = −4j + t (6i + 3j) O(x, y) = (0, 4) + t (6, 3) ai + bj = 4j + t (6i + 3j) ○ (x, y) = (0, 6) + t ( − 4, 3) ai + bj - 4j + t (6i + 3j) - (x, y) = (-4, 0) + t (3, 6) ai + bj = − 4i + t (3i + 6j) (x, y) = (6, 3) + t(0, − 4) ai + bj = (6i + 3j) + t ( − 4j)
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Let u € R³ and W = span ((1, 1, 0)", (1, -1,0)). With respect to the standard inner product, which of the following is the orthogonal projection of (3, 4, 5)¹ € R³ on W? 0 00 (1,-1,0) (1,-1)" (3,4) (3,4,0)