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If v is a vector in R", and if k is any scalar, then: (a) ||v|| 20 (h) x = 0 if and only if y = 0

If v is a vector in R", and if k is any scalar, then: (a) ||v|| 20 (h) x = 0 if and only if y = 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.6: Quadratic Functions
Problem 37E
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(Chapter3.2) Part 33, on paper please
Our first theorem in this section will generalize to R" the follo about vectors in R2 and R³: ● Distances are nonnegative. ● The zero vector is the only vector of length zero. Multiplying a vector by a scalar multiplies its length by the scalar. It is important to recognize that just because these results hold in antee that they hold in R"-their validity in R" must be proved us of n-tuples. Theorem 3.2.1 If vis a vector in R", and if k is any scalar, then: (a) ||v|| 20 (b) ||v|| = 0 if and only if v = 0 (c) ||kv|| = |k|||v|| We will prove part (c) and leave (a) and (b) as exercises. +(kun)² Proof (c) If v = (U₁, U2,..., Un), then kv = (ku₁, ku₂,..., kun), so ||kv|| = √(kv₂)² + (kv₂)² + ... (K²) (v² + v² + + Un)
Transcribed Image Text:Our first theorem in this section will generalize to R" the follo about vectors in R2 and R³: ● Distances are nonnegative. ● The zero vector is the only vector of length zero. Multiplying a vector by a scalar multiplies its length by the scalar. It is important to recognize that just because these results hold in antee that they hold in R"-their validity in R" must be proved us of n-tuples. Theorem 3.2.1 If vis a vector in R", and if k is any scalar, then: (a) ||v|| 20 (b) ||v|| = 0 if and only if v = 0 (c) ||kv|| = |k|||v|| We will prove part (c) and leave (a) and (b) as exercises. +(kun)² Proof (c) If v = (U₁, U2,..., Un), then kv = (ku₁, ku₂,..., kun), so ||kv|| = √(kv₂)² + (kv₂)² + ... (K²) (v² + v² + + Un)
10 lb 8 lb R 75% 45° FIGURE Ex-31 FIGURE Ex-32 Working with Proofs 33. Prove parts (a) and (b) of Theorem 3.2.1. 34. Prove parts (a) and (c) of Theorem 3.2.3. 35. Prove parts (d) and (e) of Theorem 3.2.3. True-False Exercises TF. In parts (a)-(j) determine whether the statem false, and justify your answer. 60° 120 N 15 100
Transcribed Image Text:10 lb 8 lb R 75% 45° FIGURE Ex-31 FIGURE Ex-32 Working with Proofs 33. Prove parts (a) and (b) of Theorem 3.2.1. 34. Prove parts (a) and (c) of Theorem 3.2.3. 35. Prove parts (d) and (e) of Theorem 3.2.3. True-False Exercises TF. In parts (a)-(j) determine whether the statem false, and justify your answer. 60° 120 N 15 100
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