12-3 - b onto a is<-1-2 2> < 3 3 4> |<-1-2 2>...

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Solutions to Homework Assignment 3 MATH 249-01 and -02 Section 12.3, Page 812 1-20, 23, 24, 26, 27, 35-40, 45, 47, 53, 58 4. < 1 / 2 , 4 > · < - 8 , - 3 > = (1 / 2)( - 8) + 4( - 3) = - 16 . 8. (4 ˆ j - 3 ˆ k ) · (2 ˆ i + 4 ˆ j + 6 ˆ k ) = 0(2) + 4(4) + ( - 3)(6) = - 2 . 14. The dot product gives the vendor’s total income that day from hamburgers, hot dogs, and soft drinks. 16. < sqrt 3 , 1 > · < 0 , 5 > = 3(0) + 1(5) = 5 . | a | = q ( 3) 2 + 1 2 = 2 and | b | = 0 2 + 5 2 = 5 . Thus a · b = 5 = (2)(5) cos θ, so cos θ = 1 2 . Therefore, θ = 60 degrees. 24. (a) Since u = - 3 4 v, u and v are parallel. (b) u · v = 1(2) + ( - 1)( - 1) + (2)(1) = 5 . This u and v are neither parallel nor perpendicular. (c) u · v = a ( - b ) + b ( a ) + c (0) = 0 , so u and v are perpendicular. 38. The scalar projection of
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Unformatted text preview: b onto a is <-1 ,-2 , 2 > · < 3 , 3 , 4 > | <-1 ,-2 , 2 > | =-1 3 . The vector projection of b onto a is therefore-1 3 <-1 ,-2 , 2 > 3 = 1 9 <-1 ,-2 , 2 > . 58. (a) The the length of any side of any triangle is less than or equal to the sum of the lengths of the other two sides. (b) | a + b | 2 = ( a + b ) · ( a + b ) = a · a + a · b + b · a + b · b = | a | 2 + 2 | a || b | cos θ + | b | 2 ≤ | a | 2 + 2 | a || b | + | b | 2 = ( | a | + | b | ) 2 . Therefore, | a + b | ≤ | a | + | b | ....
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