Explanation Given: α = v − [ v ⋅ u ‖ u ‖ 2 ] u and β = w − [ w ⋅ u ‖ u ‖ 2 ] u − [ w ⋅ α ‖ α ‖ 2 ] α Proof: Consider three nonzero vectors u → , v → and w → in R 3 that do not lie in the same plane. Two vectors α → and β → are defined as follows: α → = v → − [ v → ⋅ u → ‖ u → ‖ 2 ] u → And, β → = w → − [ w → ⋅ u → ‖ u → ‖ 2 ] u → − [ w → ⋅ α → ‖ α → ‖ 2 ] α → Two nonzero vectors are said to be orthogonal if their dot product is equal to '0.' Compute the dot product of vectors u → and α → as follows: u → ⋅ α → = u → ⋅ [ v → − { v → ⋅ u → ‖ u → ‖ 2 } u → ] = u → ⋅ v → − u → ⋅ [ { v → ⋅ u → ‖ u → ‖ 2 } ] u → = u → ⋅ v → − { v → ⋅ u → ‖ u → ‖ 2 } ( u → ⋅ u → ) = u → ⋅ v → − { v → ⋅ u → ‖ u → ‖ 2 } ‖ u → ‖ 2 Simplify further, u → ⋅ α → = u → ⋅ v → − v → ⋅ u → = u → ⋅ v → − u → ⋅ v → = 0 Therefore, "vectors u → and α → are orthogonal." Compute the dot product of vectors u → and β → as follows: As vectors u → and α → are orthogonal, substitute u → ⋅ α → = 0 u → ⋅ β → = u → ⋅ w → − ( w → ⋅ u → ‖ u → ‖ 2 ) ‖ u → ‖ 2 − ( w → ⋅ α → ‖ α → ‖ 2 ) × 0 = u → ⋅ w → − w → ⋅ u → − 0 = u → ⋅ w → − u → ⋅ w → − 0 = 0 Therefore, "vectors u → and β → are orthogonal

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 9, Problem 92SP

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To show: u,α,β are mutually orthogonal

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Chapter 9, Problem 91SP

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1. True/False:Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample. (a) True or False: For any two vectors u and v in R³, u x v = v x u. (b) True or False: If u and v are two vectors in R3, then u x v=u· v. (c) True or False: If u xv = vxu, then u and v are parallel. (d) True or False: If u, v, and w are vectors in R³, then (u x v) x w = u x (v x w). (e) True or False: The triple scalar product can be used to find the volume of a parallelepiped. (f) True or False: If u, v, and w are vectors in R³, then u · (v x w) = -v · (u x w). (g) True or False: If u and v are nonparallel vectors in R³, u. v = cot 0, where 0 is the angle between then ||ux || u and v. (h) True or False: If u and v are unit vectors in R³, then u x v is also a unit vector.

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Chapter 9 Solutions

Calculus

Ch. 9.1 - Prob. 1PS Ch. 9.1 - Prob. 2PS Ch. 9.1 - Prob. 3PS Ch. 9.1 - Prob. 4PS Ch. 9.1 - Prob. 5PS Ch. 9.1 - Prob. 6PS Ch. 9.1 - Prob. 7PS Ch. 9.1 - Prob. 8PS Ch. 9.1 - Prob. 9PS Ch. 9.1 - Prob. 10PS

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u , α , β are mutually orthogonal

Solution Summary: The author explains that u,alpha, beta are mutually orthogonal.

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