010.103-2009-1-final - ; o > 1 e K (2009 6 Z 4 6...

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Unformatted text preview: ; o > 1 e K (2009 6 Z 4 6 { 9 1:00-3:00) < : s 2 : M % K V + I U c l a Z k k . ( @ 200 ) Problem 1 (15pts) . Find the minimal distance between the origin and all points which satisfy two equations x + y + z + w = 2 , 2 x + y + 2 z + w = 3 in R 4 . Problem 2 (20pts) . Suppose that the light beamed from the origin in the direction of v = (1 , 2 , 3) is reflected onto the plane containing three points P = (1 , 2 , 3) ,Q = (2 , 3 , 1) ,R = (3 , 1 , 2) , in 3-dimensional space . Find the point at which the reflected light meets xy-plane. Problem 3 (30pts) . Let u and v be mutually perpendicular unit vectors in R 3 and H be the plane containing u and v . Answer the following questions. (Here, the initial point of u and v is the origin.) (a) Show that for a vector a in R 3 , the closest vector to a on H is ( a · u ) u + ( a · v ) v . (b) The map P : R 3 → R 3 is defined by P ( a ) = ( a · u ) u + ( a · v ) v . Show that P is a linear transformation....
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This note was uploaded on 09/11/2009 for the course MATHMATICS 010-102 taught by Professor Cho during the Spring '09 term at Seoul National.

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