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31. Prove parts (a), (c), and (d) of Theorem 3.1.1.

31. Prove parts (a), (c), and (d) of Theorem 3.1.1.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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FIGURE Ex-29 30. What is the sum of all radial vectors of a regular n-side gon? (See Figure Ex-29.) Working with Proofs 31. Prove parts (a), (c), and (d) of Theorem 3.1.1. 32. Prove parts (e)-(h) of Theorem 3.1.1. 33. Prove parts (a)-(c) of Theorem 3.1.2.
Transcribed Image Text:FIGURE Ex-29 30. What is the sum of all radial vectors of a regular n-side gon? (See Figure Ex-29.) Working with Proofs 31. Prove parts (a), (c), and (d) of Theorem 3.1.1. 32. Prove parts (e)-(h) of Theorem 3.1.1. 33. Prove parts (a)-(c) of Theorem 3.1.2.
EXAMPLE 3 | Algebraic Operations Using Components If v = (1, -3, 2) and w = (4, 2, 1), then v+w=(5,-1, 3), 2v = (2,-6,4) -w = (-4,-2, -1), v-w=v+(-w) = (-3, -5,1) The following theorem summarizes the most important properties of ve tions. Theorem 3.1.1 If u, v, and w are vectors in R", and if k and m are scalars, then: (a) u + v=v+u (b) (u+v)+w=u+ (v + w) (c) u+0=0+u=u (d) u + (-u) = 0 (e) k(u + v) = ku + kv (f) (k+m)u = ku + mu (g) k(mu) = (km)u (h) lu= u We will prove part (b) and leave some of the other proofs as exercises. Proof (b) Let u = (U₁, U₂, ..., Un), V = (U₁, U₂, ..., Un), and w = (W1, W2,...,U (u + v) + w = ((U₁, U₂,..., Un) + (V₁, V₂,..., Un)) + (W₁, W₂,..., wn) [Vect = = (U₁ + V₁, U₂+U2,..., Un + Un) + (W₁, W2,..., wn) = ((U₁ + v₁) + w₁, (u₂ + V₂) + W2,..., .., (un + vn) + wn) [Vect [Regr = (u₁ + (v₁ + w₁), U₂ + (V₂ + W₂),..., Un + (Un+wn))
Transcribed Image Text:EXAMPLE 3 | Algebraic Operations Using Components If v = (1, -3, 2) and w = (4, 2, 1), then v+w=(5,-1, 3), 2v = (2,-6,4) -w = (-4,-2, -1), v-w=v+(-w) = (-3, -5,1) The following theorem summarizes the most important properties of ve tions. Theorem 3.1.1 If u, v, and w are vectors in R", and if k and m are scalars, then: (a) u + v=v+u (b) (u+v)+w=u+ (v + w) (c) u+0=0+u=u (d) u + (-u) = 0 (e) k(u + v) = ku + kv (f) (k+m)u = ku + mu (g) k(mu) = (km)u (h) lu= u We will prove part (b) and leave some of the other proofs as exercises. Proof (b) Let u = (U₁, U₂, ..., Un), V = (U₁, U₂, ..., Un), and w = (W1, W2,...,U (u + v) + w = ((U₁, U₂,..., Un) + (V₁, V₂,..., Un)) + (W₁, W₂,..., wn) [Vect = = (U₁ + V₁, U₂+U2,..., Un + Un) + (W₁, W2,..., wn) = ((U₁ + v₁) + w₁, (u₂ + V₂) + W2,..., .., (un + vn) + wn) [Vect [Regr = (u₁ + (v₁ + w₁), U₂ + (V₂ + W₂),..., Un + (Un+wn))
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