lina-09-10-09-p1

# Lina-p1 - a,b,c> 0 and angle γ between legs a and b see the ﬁgure below We want to prove the law of cosines c 2 = a 2 b 2-2 ab cos γ(1 and

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Linear Algebra Geometry: Sheet 1 Set on Friday, October 9: Question 1 (a) (b) (c), Question 3, Question 6 (a) (d) and Question 7 1. Sketch the following vectors in R 2 and compute their norm k v k (a) v 1 = ± 2 5 ² (b) v 2 = ± - 2 5 ² (c) v 3 = ± 0 2 ² (d) v 4 = ± 0 - 2 ² (e) v 5 = ± - 1 - 5 ² (f) v 6 = ± 0 0 ² 2. Find the components of (a) 5 v 1 (b) v 2 + v 1 (c) 5 2 v 3 - v 2 (d) 5( v 4 + 2 v 2 ) (e) 2 v 5 + v 1 + v 2 (f) 0 v 1 where v 1 , v 2 , v 3 , v 4 , v 5 and v 6 are the vectors from Exercise 1. 3. Let u = ± 1 2 ² , v = ± 2 - 3 ² and w = ± 3 2 ² . Find the components of the vector x R 2 that satisﬁes 2 u - v - x = 7 x + w . 4. Let u = ± 1 0 ² and v = ± 1 y ² with y R . Compute k u k , k v k and k u + v k , and determine for which y R we have k u + v k = k u k + k v k . Sketch the vectors u , v and u + v in this case. 5. For u , v R 2 show that k u - v k ≤ k u k + k v k , and give an example of a u and v for which k u - v k ≤ k u k + k v k . 6. Compute the distances between the following points in R 2 (a) ± 2 5 ² and ± - 1 0 ² (b) ± 10 2 ² and ± 11 4 ² (c) ± 0 1 ² and ± 1 0 ² (d) 3 ± 0 - 2 ² and 5 ± 0 - 2 ² (e) ± cos ϕ sin ϕ ² and ± 1 0 ² 1

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7. We are given a triangle with sidelength
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Unformatted text preview: a,b,c > 0 and angle γ between legs a and b , see the ﬁgure below. We want to prove the law of cosines c 2 = a 2 + b 2-2 ab cos γ , (1) and derive the triangle inequality. (i) Consider the vectors u = ± a ² and v = b ± cos γ sin γ ² . Show that k u k = a and k v k = b . (ii) Plot the vectors u and v and show that they span the triangle with sides a,b,c . Conclude that c = k u-v k and use this to derive (1). (iii) Use the law of cosines to derive the triangle inequality in the form c 2 ≤ ( a + b ) 2 , and determine for which γ ∈ [0 ,π ] we have equality. a b c γ 2...
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## This note was uploaded on 02/25/2010 for the course MATHEMATIC 11007 taught by Professor Spyros during the Spring '10 term at Bristol Community College.

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Lina-p1 - a,b,c> 0 and angle γ between legs a and b see the ﬁgure below We want to prove the law of cosines c 2 = a 2 b 2-2 ab cos γ(1 and

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