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Unformatted text preview: SOLUTIONS TO ASSIGNMENT #4 Math 152 1. Consider the intersection between the following two planes given in parametric form: P 1 : x = [2 , 4 , 3] + s 1 [1 , 2 , 1] + s 2 [2 , 5 , 4] P 2 : x = [1 , , 5] + t 1 [3 , 8 , 7] + t 2 [2 , 1 , 5] Find the intersection as a line in parametric form. Solution: We will solve the equation x = [2 , 4 , 3] + s 1 [1 , 2 , 1] + s 2 [2 , 5 , 4] = [1 , , 5] + t 1 [3 , 8 , 7] + t 2 [2 , 1 , 5] or s 1 [1 , 2 , 1] + s 2 [2 , 5 , 4] t 1 [3 , 8 , 7] t 2 [2 , 1 , 5] = [ 1 , 4 , 8] for s 1 , s 2 , t 1 and t 2 . To solve, we write the system as an augmented matrix and reduce it to rowechelon form. 1 2 3 2 1 2 5 8 1 4 1 4 7 5 8 1 2 3 2 1 0 1 2 3 2 0 2 4 7 7 R 2 R 2 2 R 1 R 3 R 3 R 1 1 2 3 2 1 0 1 2 3 2 0 0 1 3 R 3 R 3 2 R 2 Therefore, t 1 (the 3rd variable) is a free variable and by back substitution, t 2 = 3 s 2 2 t 1 + 3 t 2 = 2 s 2 = 7 + 2 t 1 s 1 + 2 s 2 3 t 1 2 t 2 = 1 s 1 = 21 t 1 We actually do not need s 1 and s 2 because we can just plug t 2 = 3 into P 2 and get the equation of the line x = [1 , , 5] + t 1 [3 , 8 , 7] 3[2 , 1 , 5] or x = [ 5 , 3 , 10] + t 1 [3 , 8 , 7] 2. Reduce the matrix  1 2 0 1 4 1 2 0 0 1 2 6 2 4 to reduced row echelon form and determine the rank of the matrix. Solution:  1 2 0 1 4 1 2 0 0 1 2 6 2 4  1 2 0 1 4 0 1 3 2 2 6 8 R 2 R 2 + R 1 R 3 R 3 + 2 R 1  1 2 0 1 4 2 2 6 8 0 1 3 R 2 R 3...
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This note was uploaded on 03/30/2011 for the course MATH 152 taught by Professor Caddmen during the Spring '08 term at The University of British Columbia.
 Spring '08
 Caddmen
 Math

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