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
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.3: Vectors
Problem 11E
Related questions
Question
![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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb7e19683-a2f9-4d14-9773-6a59da469282%2F3a055573-7a70-4d16-8212-8a85af08fb2b%2Fhdqb6ya_processed.jpeg&w=3840&q=75)
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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