lab2_sol_f_08revised2

# lab2_sol_f_08revised2 - Math 115 Lab 2 Fall 2008 Topics •...

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Unformatted text preview: Math 115 - Lab 2 - Fall 2008. Topics: • Projections and planes • Distance of a point to a line or a plane in R 3 • Solutions and elementary operations to linear systems, • Gaussian elimination, • Homogeneous equations You are to provide full solutions to the following problems. You are allowed, and encouraged to collaborate with your classmates, use your notes and textbook and ask the TA for guidance. Direct copying of solutions is not encouraged, nor is it allowed or ethical. Last name: First name: Student number: Math 115 - Lab 2 - Fall 2008. Student number: 1. Let y = (- 1 ,- 2 , 2) and x = (1 ,- 2 , 1). a) Find the projection proj x y of the vector y onto the vector x . Solution: proj x y = < x , y > x bardbl x bardbl 2 = < (- 1 ,- 2 , 2) , (1 ,- 2 , 1) > (1 ,- 2 , 1) 6 = 5 6 (1 ,- 2 , 1) b) Find the point on the line L : x = (2 , 2 ,- 1) + t (1 ,- 2 , 1) which is closest to y = (1 , , 1). Solution: • Let a = (2 , 2 ,- 1) and d = (1 ,- 2 , 1) (the direction vector of the line L ). • We want the point proj d ( y- a ) + a . proj d ( y- a ) + a = < (1 , , 1)- (2 , 2 ,- 1) , (1 ,- 2 , 1) > (1 ,- 2 , 1) bardbl (1 ,- 2 , 1) bardbl 2 + (2 , 2 ,- 1) = < (- 1 ,- 2 , 2) , (1 ,- 2 , 1) > (1 ,- 2 , 1) 6 + (2 , 2 ,- 1) = 5 6 (1 ,- 2 , 1) + (2 , 2 ,- 1) = parenleftbigg 17 6 , 1 3 ,- 1 6 parenrightbigg c) Find the minimum distance of the point y = (1 , , 1) to the line L : x = (2 , 2 ,- 1)+ t (1 ,- 2 , 1)....
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lab2_sol_f_08revised2 - Math 115 Lab 2 Fall 2008 Topics •...

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