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1. Using ā = (a₁, A2, A3), b = (b₁,b₂, b3), C = (C₁, C₂, C3) and k as scalar, prove all properties of dot product as listed below. i. ā · ā = |ā|² ii. ā ⋅ b = b ·ā iii. ā. (b + c) = ā·b+ā·č iv. (kā) · b = k(ā · b) = ā · (kb) v. Ō ·ā = 0 2. Using ā = (a₁, A2, A3), b = (b₁,b₂, bz), C = (C₁, C₂, C3) and k as scalar, prove all properties of cross product as listed below. i. āx b = -bxā ii. (kā) × b = k(āx b) = a × (kb) iii. āx (b + c) = (axb) + (ax c) iv. (ā + b) × c = (āx c) + (b × c) v. ā · (b × c) = (ā × b) · c - vi. ā × (b × c) = (ā · c)b − (ā · b)c

1. Using ā = (a₁, A2, A3), b = (b₁,b₂, b3), C = (C₁, C₂, C3) and k as scalar, prove all properties of dot product as listed below. i. ā · ā = |ā|² ii. ā ⋅ b = b ·ā iii. ā. (b + c) = ā·b+ā·č iv. (kā) · b = k(ā · b) = ā · (kb) v. Ō ·ā = 0 2. Using ā = (a₁, A2, A3), b = (b₁,b₂, bz), C = (C₁, C₂, C3) and k as scalar, prove all properties of cross product as listed below. i. āx b = -bxā ii. (kā) × b = k(āx b) = a × (kb) iii. āx (b + c) = (axb) + (ax c) iv. (ā + b) × c = (āx c) + (b × c) v. ā · (b × c) = (ā × b) · c - vi. ā × (b × c) = (ā · c)b − (ā · b)c

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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1. Using a = (a₁, A₂, A3), b = (b₁,b₂, B3), C = (C₁, C2, C3) and k as scalar, prove all properties of dot product as listed below. i. ā.ā = Tā|² ii. ā.b = b.a : a iii. ā. (b + c) = ā·b + ā·č iv. (kā) b = k(ā b) =ā. (kb) v. Ō ·ā = 0 2. Using a = (a₁, α₂, α3), b = (b₁,b₂, bz), = (C₁, C₂, C3) and k as scalar, prove all properties of cross product as listed below. i. āx b-bx ā ii. (kā) × b = k(ā × b) = ā× (kb) iii. a x (b + c) = (ā x b) + (ā xã iv. (a + b) x c = (ax c) + (bx c) (ā v. ā. (bx c) = (a x b).c - vi. ā × (b × c) = (ā· c)b − (ā· b)c
Transcribed Image Text:1. Using a = (a₁, A₂, A3), b = (b₁,b₂, B3), C = (C₁, C2, C3) and k as scalar, prove all properties of dot product as listed below. i. ā.ā = Tā|² ii. ā.b = b.a : a iii. ā. (b + c) = ā·b + ā·č iv. (kā) b = k(ā b) =ā. (kb) v. Ō ·ā = 0 2. Using a = (a₁, α₂, α3), b = (b₁,b₂, bz), = (C₁, C₂, C3) and k as scalar, prove all properties of cross product as listed below. i. āx b-bx ā ii. (kā) × b = k(ā × b) = ā× (kb) iii. a x (b + c) = (ā x b) + (ā xã iv. (a + b) x c = (ax c) + (bx c) (ā v. ā. (bx c) = (a x b).c - vi. ā × (b × c) = (ā· c)b − (ā· b)c
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