1211a1 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 1st Semester: Assignment 1 Due date: 2010 September 17 (Friday), 5:00 pm. Please hand in problem 1-6 for assignment one only and please leave problem 7, 8, 9 to assignment 2. 1. Let a , b , c be vectors in R 3 . Verify the following equalities: (a) ( a × b ) × c = ( a · c ) b - ( b · c ) a . (b) ( a × b ) · ( c × d ) = ( a · c )( b · d ) - ( a · d )( b · c ). (c) ( a × b ) × c + ( b × c ) × a + ( c × a ) × b = 0 . (This is known as the Jacobi identity.) 2. Suppose that l 1 ( t ) = t a + b 1 and l 2 ( t ) = t a + b 2 are parallel lines in R 3 . Show that the distance D between them is given by D = k a × ( b 2 - b 1 ) k k a k . (Hint: Consider Example 7 on Page 44.) 3. (a) Show that the distance d between the two parallel planes determined by the equations Ax + By + Cz = D 1 and Ax + By + Cz = D 2 is d = | D 1 - D 2 | A 2 + B 2 + C 2 . (b) Write equations for the planes that are parallel to
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This note was uploaded on 05/04/2011 for the course MATH 1211 taught by Professor Wang during the Spring '11 term at HKU.

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1211a1 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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