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Unformatted text preview: Math 136 Assignment 1 Solutions 1. Compute each of the following. a) (1 , 3 , 4) + ( 1 , 1 , 2) Solution: (1 , 3 , 4) + ( 1 , 1 , 2) = (0 , 4 , 6). b) 3( 1 , 1 , 2) 2(2 , , 3). Solution: 3( 1 , 1 , 2) 2(2 , , 3) = ( 3 , 3 , 6) + ( 4 , , 6) = ( 7 , 3 , 12). 2. Determine the distance between P (2 , 1 , 1) and Q (1 , 1 , 1). Solution: The distance is k (1 , 1 , 1) (2 , 1 , 1) k = k ( 1 , 2 , 0) k = p ( 1) 2 + ( 2) 2 + 0 2 = 5. 3. Determine which of the following pairs of vectors is orthogonal. a) (2 , 1 , 1), ( 2 , 1 , 3). Solution: We have (2 , 1 , 1) ( 2 , 1 , 3) = 4 + 1 + 3 = 0 so they are orthogonal. b) ( 1 , 3 , 6), (3 , 1 , 0). Solution: We have ( 1 , 3 , 6) (3 , 1 , 0) = 3 3 + 0 = 6 so they are not orthogonal. 4. Find an equation for the plane through P (3 , 2 , 1) and parallel to x 1 x 2 + 2 x 3 = 4. Solution: Since the plane is parallel to x 1 x 2 + 2 x 3 = 4 it must have normal vector ~n = (1 , 1 , 2). Thus the equation of the plane is x 1 x 2 + 2 x 3 = 1(3) 1( 2) + 2(1) = 7 . 5. For each of the following sets: i) Determine if the set is linearly dependent or linearly independent. Justify....
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 Spring '08
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 Math, Linear Algebra, Algebra

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