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math226fall11mt1_3s

# math226fall11mt1_3s - The plane passes through Q and is...

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because u · P ~ Q = 3 , | ~u | = 6 , comp ~u P ~ Q = ~u · P ~ Q | ~u | = 3 6 . 5. (a) Find a direction vector of the intersection line of two planes x + y = 2 z and 2 x + z = 10. Answer. We take orthogonal vectors to both planes ( ~a is orthogonal to the ﬁrst plane and ~ b to the second): ~a = h 1 , 1 , - 2 i , ~ b = h 2 , 0 , 1 i . Then ~u = ~a × ~ b is a direction vector of the intersection line. We ﬁnd ~u = ± ± ± ± ± ± ± ~ i ~ j ~ k 1 1 - 2 2 0 1 ± ± ± ± ± ± ± = ~ i - 5 ~ j - 2 ~ k. (b) Write an equation of the plane through Q (5 , 2 , 1) that is perpendicular to two planes x + y = 2 z and 2 x + z = 10. Find the distance from Q to the plane 2 x + z = 10 . Answer.
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Unformatted text preview: The plane passes through Q and is orthogonal to the intersection line with a direction vector ~u : ( x-5)-5( y-2)-2( z-1) = 0 or x-5 y-2 z + 7 = 0 . To ﬁnd the distance we choose a point on 2 x + z = 10: for example, P (5 , , 0). Then P ~ Q = h , 2 , 1 i ,P ~ Q · ~ b = 1 and the distance d = | comp ~ b P ~ Q | = 1 √ 5 , where ~ b = h 2 , , 1 i is an orthogonal vector to 2 x + z = 10. 3...
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math226fall11mt1_3s - The plane passes through Q and is...

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