midterm 2006(1) ans

midterm 2006(1) ans - HKUST MATH 102 Midterm Examination...

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Unformatted text preview: HKUST MATH 102 Midterm Examination Multivariable and Vector Calculus 6 Nov 2006 Answer ALL 5 questions Time allowed – 120 minutes Directions – This is a closed book examination. No talking or whispering are allowed. Work must be shown to receive points. An answer alone is not enough. Please write neatly. Answers which are illegible for the grader cannot be given credit. Note that you can work on both sides of the paper and do not detach pages from this exam packet or unstaple the packet. Student Name: Student Number: Tutorial Session: Question No. Marks 1 /20 2 /20 3 /20 4 /20 5 /20 Total /100 – 1 – Problem 1 (20 points) Your Score: (a) Assume a , b and c are three dimensional vectors and if ( a × b ) × c = λ a + μ b + β c . Use suffix notation to find λ , μ and β in terms of the vectors a , b and c . Can you say something about the direction of the vector ( a × b ) × c . (b) (i) Find the distance (in terms of n , r and r 1 only) from the point r 1 to the plane ( r- r ) · n = 0. (ii) Use (i) or otherwise, find the distance d between the two parallel planes determined by the equations Ax + By + Cz = D 1 and Ax + By + Cz = D 2 . (iii) Use (ii) or otherwise, find equations for the planes that are parallel to x + 3 y- 5 z = 2 and lie three units from it. Solution: (a) [( a × b ) × c ] i = ² ijk ( a × b ) j c k = ² ijk ² jpq a p b q c k = ² jki ² jpq a p b q c k = ( δ kp δ iq- δ kq δ ip ) a p b q c k = a k b i c k- a i b k c k i.e. ( a × b ) × c = ( a · c ) b- ( b · c ) a , i.e. μ = a · c , λ =- b · c and β = 0. The resulting vector of ( a × b ) × c is a linear combination of the vectors a and b , hence it lies on the plane containing the vectors a and b ....
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midterm 2006(1) ans - HKUST MATH 102 Midterm Examination...

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