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Exercise 4.12. For the pairs of vectors v, w below, compute projwv and proj,w. Also, verify...
In Exercises 78-79, the given vectors are respect to the Euclidean inner product. Find proj (1, 2, 0, -2) and W is the subspace of R4 spanned by the orthogonal with X, where vectors. v Stios 78. (a) Vi (1, 1, 1, 1), v2 = (-1, -1, -1, 1) (b) vi = (1,0, -3, -1), v2 = (4, 2, 1, 1)
In Exercises 78-79, the given vectors are respect to the Euclidean inner product. Find proj (1, 2, 0, -2)...
Given v = 10i - 4j and w=1- ), a. Find projwv. b. Decompose v into two vectors V, and V2, where V, is parallel to wand v2 is orthogonal to w. a. projwv= (Type your answer in terms of i andj.) b. Vo = (Type your answer in terms of i and j.) V2 (Type your answer in terms of i andj.)
2. Which of the following pairs of vectors are orthogonal? (a) v = 3i - 2j, w = --i +2j (b) v = -2i, w = 5j (c) v = -i + 2j, w = -1 (d) v = 2i – 3j, w = -2i + 3j (e) None of these
Problem #5: (a) Let u = (10, 4, -1, 8) and v = (-5, 10, -6, -1). Find ||u - proj,u||. Note: You can partially check your work by first calculating proj^u, and then verifying that the vectors projyu and u proj^u are orthogonal (b) Consider the following vectors u, v, w, and z (which you can copy and paste directly into Matlab). u -8.9 -9.8 -5.8], v = [0.8-4.1 -3.71, w = [8.6 -9.1 -8.11, [3.4 4.8 3.11 z...
Decompose v into two vectors, V, and v2, where v, is parallel to w and v2 is orthogonal to w. V= -1 + 2), w=i+2) V1 = i + V2 = ((i+O; (Simplify your answer.)
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...
7. The set {u, v, w} is an orthogonal set of vectors, where u= (0,3,4), v = (1,0,0) and w = (0,4, -3). If (0,-1,-1) = au + bu + cw, then (a, b, c) = mark (x) the correct answer: A (-3,0,-) B (-2, 0, - 2) C (7,0, ) D(-2,0, 35) E (-7,0, -1) F (0,-1, -1)
Decompose v into two vectors, V, and V2, where vais parallel to w and v2 is orthogonal to w. V= -1 -2], w = -21 - v1 = ( Oi+O; v2 = (i+O; (Simplify your answer.)
3. Given pairwise orthogonal vectors u, v, w ER(each vector is orthogonal to every other), with || || = ||0|| = ||w|| = 1, and C1, C2, C3 € R, prove that || Cu + c2v + c3w||2 = cſ + cx+cz.
1. (10 points) Consider the vectors u = 0 and v = | 2 [E (a) Find cosine of the angle between two vectors. Is the angle acute, obtuse, or neither? (b) Find p = projspan{v}u and verify that u-p is orthogonal to v.