MATH311ansHW1

MATH311ansHW1 - 150) (153 , 129 , 150) feet. 1.3.3. x = (1...

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Math 311, Homework 1 partial answers and solutions 1.1.9. We need to solve the system 3 a = 9 , - a + b = - 1 , 5 b = 10 . There is exactly one solution ( a = 3 , b = 2) . 1.1.12. Answer: the only vector with the first coordinate 6 that is a linear combination of u and v is (6 , - 2 , 4) . 1.1.14. Answers: (a) i . (b) j . (c) k . 1.2.25. For t 0 , | (1 - t ) x + t y - x | = | x - t x + t y - x | = | t ( y - x ) | = | t || y - x | = t | y - x | The second part is similar, and uses (1 - t ) 0 . If t is not between 0 and 1 , the conclusion is false. For example, if x = (0 , 0) , y = (0 , 1) , and t = - 1 , then | ((1 - t ) x + t y ) - x | = 1 while t | y - x | = - 1 . Geometrically, the condition 0 t 1 guarantees that (1 - t ) x + t y lies on the segment between x and y . 1.2.30. (a) Answer: the midpoints are (5 / 2) i and i +2 j , one of two vectors connecting them is (3 / 2) i - 2 j , which is half of the vector 3 i - 4 j representing the third side of the triangle. (b) 1 2 b - 1 2 a = 1 2 ( b - a ) . 1.2.32. Answer: (200 cos 40 , 200 sin 40 ,
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Unformatted text preview: 150) (153 , 129 , 150) feet. 1.3.3. x = (1 , 2 , 2) + t (1 , , 1) . 1.3.5. (a) See the textbook. (b) If p = t a + b = s c + d , then p 1 =-t = 2 s-3 , p 2 = t + 1 = s + 2 . Adding the equations, 1 = 3 s-1 , so s = 2 3 and t = 5 3 . We check that 5 3 a + b = (-5 3 , 8 3 ) = 2 3 c + d . 1 (c) If c = (2 ,-2) , then c =-2 a , so the lines are parallel, and since they do not coincide, they do not intersect. Algebraically, we now get equations p 1 =-t = 2 s-3 , p 2 = t + 1 =-2 s + 2 , adding which produces 1 =-1 , which is obviously false for any s, t . 1.3.13. Linearly independent. 1.3.14. Not linearly independent....
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MATH311ansHW1 - 150) (153 , 129 , 150) feet. 1.3.3. x = (1...

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