pset1sol - 18.06 Problem Set 1 Solutions Problem 1 Do...

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18.06 Problem Set 1 Solutions Problem 1: Do problem 27 from section 1.2 in the book. Solution (10pts) || v - w || ≤ || v || + || w || = 5 + 3 = 8 and || v - w || ≥ || v || - || w || = 5 - 3 = 2 . (5pts) | v · w | = || v || · || w || cos θ ≤ || v || · || w || Thus we find that -|| v |||| w || ≤ | v · w | ≤ || v || · || w || . Thus the minimum value occurs when the dot product is a small as possible: ie. v and w are parrallel, but point in opposite directions. So smallest value is -15. The maximum value occurs when the dot product is as large as possible, thus occurs when v and w are parallel and point in the same direction. Thus the largest value is 15. (5pts) Problem 2: Do problem 8 from section 2.1. Solution (10pts) Normally 4 ”planes” in 4-dimensional space meet at a point (2pts). Normally 4 column vectors in a 4-dimensional space can combine to produce b . The combination of the 4 column vectors producing b is: 1 1 1 1 0 + 2 1 1 1 1 = 3 3 3 2 (4pts) The system of linear equations this is satisfying is (4pts): x + y + z + t = 3 y + z + t = 3 z + t = 3 t = 2 .
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