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# exam1-solution - Math233 Exam 1 Fall 2006 Problem 1 Let A...

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Math233 Exam 1 Fall 2006 Problem 1. Let A = ( x, 2 x, 0) , B = (0 , 4 , 2) , and C = (2 , 0 , - 1) . Find x so that the triangle ABC is a right triangle at B . ABC is a right triangle at B if and only if ~ AB · ~ BC = 0 . ~ AB · ~ BC = ( - x, 4 - 2 x, 2) · (2 , - 4 , - 3) = - 2 x - 4(4 - 2 x ) - 6 = 6 x - 22 So 6 x - 22 = 0 , which implies that x = 11 / 3 . Problem 2. Give an example of equations of two planes P 1 and P 2 satisfying the following conditions: (1) P 1 and P 2 are parallel; (2) P 1 contains the origin (0 , 0 , 0) and P 2 contains the point (1 , 1 , 1) . [Answers may vary] A simple example is: P 1 : x = 0 and P 2 : x = 1 . Their respective normal vectors are ~n 1 = ~n 2 = (1 , 0 , 0) , and are clearly colinear (condition (2) is also trivially satisﬁed). Another example is: P 1 : x + y + z = 0 and P 2 : x + y + z = 3 . Problem 3. Find parametric equations and symmetric equations for the line through (1 , 2 , 3) and perpen- dicular to the plane z = 5 .

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exam1-solution - Math233 Exam 1 Fall 2006 Problem 1 Let A...

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