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Unformatted text preview: Fall 2007 ARE211 Problem Set #06 Answer key First Lin Algebra Problem Set (1) Simon & Blume question 10.5 (page 208). Ans: Let u = (1,2), v = (0,1), w = (1,3), x = (1,2,0) and z = (0,1,1) (i) u + v = (1,2)+(0,1) = (1,3). (ii) 4* w = 4*(1,3) = (4,12). (iii) u + z is not defined as u is an (1x2) vector and z is an (1x3) vector. (iv) 3* z = 3*(0,1,1) = (0,3,3). (v) 2* v = 2*(0,1) = (0,2). (vi) u +2* v = (1,2)+2*(0,1) = (1,4). (vii) u v = (1,2)(0,1) = (1,1). (viii) 3* x + z = 3*(1,2,0)+(0,1,1) = (3,7,1). (ix) 2* x = 2*(1,2,0) = (2,4,0). (x) w + 2* x is not defined as w is an (1x2) vector and x is an (1x3) vector 2 (2) Simon & Blume question 10.10 (page 220) Ans: The vectors for part a,b,d,e, and g are displayed in figure 1. The vectors for part c and f are diplayed in figure 2. Figure 1. The vectors for part a,b,d,e, and g 1 2 3 4123 1 2 3 4123 x1axis x2axis part (a) part (b) part (d) part (e) part (g) a)  (3 , 4)  = √ 3 2 + 4 2 = √ 9 + 16 = √ 25 = 5 b)  (0 , 3)  = radicalbig 2 + ( 3) 2 = √ 0 + 9 = √ 9 = 3 c)  (1 , 1 , 1)  = √ 1 2 + 1 2 + 1 2 = √ 1 + 1 + 1 = √ 3 d)  (3 , 3)  = √ 3 2 + 3 2 = √ 9 + 9 = √ 2 * 9 = 3 * √ 2 e)  ( 1 , 1)  = radicalbig ( 1) 2 + ( 1) 2 = √ 1 + 1 = √ 2 f)  (1 , 2 , 3)  = √ 1 2 + 2 2 + 3 2 = √ 1 + 4 + 9 = √ 14 g)  (2 , 0)  = √ 2 2 + 0 2 = √ 4 + 0 = √ 4 = 2 h)  (1 , 2 , 3 , 4)  = √ 1 2 + 2 2 + 3 2 + 4 2 = √ 1 + 4 + 9 + 16 = √ 30 i)  (3 , , , 0)  = √ 3 2 + 0 2 + 0 2 + 0 2 = √ 9 + 0 + 0 + 0 = √ 9 = 3 3 (3) Suppose you know that the angle between two vectors is as given below. What do you know about the sign of the inner product of the two vectors? a) 180 b) 53 c) 320 d) 90 Ans: Recall that the sign of the inner product of two vectors v 1 and v 2 is given by the sign of the cosinus of the angle between them. a) cos (180) = 1 ⇒ The sign of the inner product is negative. Figure 2. The vectors for part 2c and 2f 1 2 3 1 2 3 1 2 3 x1axis part (c) part (f) x2axis x3axis 4 b) cos (53) > ⇒ The sign of the inner product is positive. c) cos (320) > ⇒ The sign of the inner product is positive. d) cos (90) = 0 ⇒ The sign of the inner product is zero as the vectors are perpendicular....
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This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at Berkeley.
 Fall '07
 Simon

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