assignment1_sol

assignment1_sol - MATH1211/10-11(2/Asol1/TNK,SKC Department...

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MATH1211/10-11(2)/Asol1/TNK,SKC Department of Mathematics, The University of Hong Kong MATH1211 (2010–11, 2nd semester) Suggested solution to Assignment 1 Note : Some solutions given below are outline only. You may need to give more details in your solution. The parts that are typed in red color are remarks for you to read; they are not a part of the solution. 1. Note that k a × b k = k a kk b k sin(angle between a and b ) is two times the area of the triangle whose two of the sides are formed by a and b with the tails of a and b touching together to form a vertex. Hence the surface area of the tetrahedron is given by k a × b k + k c × a k + k b × c k + k ( b - a ) × ( c - a ) k 2 . 2. Let a = a 1 i + a 2 j + a 3 k , b = b 1 i + b 2 j + b 3 k and c = c 1 i + c 2 j + c 3 k . Then a × b = ( a 2 b 3 - a 3 b 2 ) i + ( a 3 b 1 - a 1 b 3 ) j + ( a 1 b 2 - a 2 b 1 ) k . Hence ( a × b ) × c = (( a 3 b 1 - a 1 b 3 ) c 3 - ( a 1 b 2 - a 2 b 1 ) c 2 ) i + (( a 1 b 2 - a 2 b 1 ) c 1 - ( a 2 b 3 - a 3 b 2 ) c 3 ) j +(( a 2 b 3 - a 3 b 2 ) c 2 - ( a 3 b 1 - a 1 b 3 ) c 1 ) k = (( a 2 c 2 + a 3 c 3 ) b 1 - ( b 2 c 2 + b 3 c 3 ) a 1 ) i + (( a 1 c 1 + a 3 c 3 ) b 2 - ( b 1 c 1 + b 3 c 3 ) a 2 ) j +(( a 1 c 1 + a 2 c 2 ) b 3 - ( b 1 c 1 + b 2 c 2 ) a 3 ) k = (( a 1 c 1 + a 2 c 2 + a 3 c 3 ) b 1 - ( b 1 c 1 + b 2 c 2 + b 3 c 3 ) a 1 ) i +(( a 1 c 1 + a 2 c 2 + a 3 c 3 ) b 2 - ( b 1 c 1 + b 2 c 2 + b 3 c 3 ) a 2 ) j +(( a 1 c 1 + a 2 c 2 + a 3 c 3 ) b 3 - ( b 1 c 1 + b 2 c 2 + b 3
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assignment1_sol - MATH1211/10-11(2/Asol1/TNK,SKC Department...

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