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2010 Algebra Test 2

# 2010 Algebra Test 2 - 1 Math1151 Algebra S1 2010 Test 2...

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1. Math1151 Algebra. S1 2010 Test 2. Version 2a. (1) Is the vector u = (1 , - 5 , - 2) T parallel to the plane x = (2 , 4 , - 1) T + λv + μw where v = (1 , 2 , 2) T and w = (3 , - 1 , 7) T ? Solution. The vector u is parallel to the plane above if u span { v, w } , that is, if there exists λ, μ such that v = λv + μw . Putting into matrix form and row reducing gives 1 3 | 1 2 - 1 | - 5 2 7 | - 2 -→ 1 3 | 1 0 - 7 | - 7 0 1 | - 4 -→ 1 3 | 1 0 - 7 | - 7 0 0 | - 5 so u = λv + μw have no solutions. Hence u span { v, w } . (2) Compute D = 1 3 - 1 2 1 0 - 1 5 7 Solution. D = 1(7) - 3(14) - (10 + 1) = - 46 (3) Find the shortest distance from the point with coordinate vector a = (3 , - 1 , 4) T to the plane Π with equation x + 3 y + 3 z = 5. Solution. We first pick a point b = ( - 1 , 1 , 1) on Π and note that n = (1 , 3 , 3) is normal to the Π. The shortest distance from a to Π is given by a - b = (4 , - 2 , 3) d = | proj n ( a - b ) | = | ( a - b ) · n | | n | = 7 19 . (4) A fifth of the items handled by a courier company are parcels, and the rest are documents. The company delivers 80% of parcels on time and delivers 90% of documents on time. If an item is selected at random, find

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2010 Algebra Test 2 - 1 Math1151 Algebra S1 2010 Test 2...

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