Assignment2sol - MAT235-CALCULUS II FALL 2010 ASSIGNMENT#2...

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MAT235-CALCULUS II, FALL 2010 ASSIGNMENT #2, Solutions Problem 1 Identify and sketch the quadric surface consisting of all points ( x, y, z ) with x - y 2 +2 y + z 2 = 2. Justify your answer carefully. It is recommended that you try to sketch with pencil so that you can improve your sketch. Scanned solution is posted onto the next file Problem 2 Let A, B, C, D be four points in R 3 which are not necessarily coplanar. Let M be the midpoint of the segment AC , and N be the midpoint of the segment BD . If --→ MB · --→ MD = --→ NA · --→ NC, show that | -→ AC | = | --→ BD | . Hint: Observe that -→ AC = --→ NC - --→ NA, and --→ BD = --→ MD - --→ MB. (1) Also show that 2 --→ MN = --→ MB + --→ MD, 2 --→ NM = --→ NA + --→ NC . Explain why the last two relations imply that | --→ MB + --→ MD | = | --→ NA + --→ NC | (2) Use (1), (2) and the assumption of the exercise to show the desired result. The identity | -→ a | 2 = -→ a · -→ a is useful. Solution : We have --→ MN = --→ MB + --→ BN, (3) and --→ MN = --→ MD + --→ DN. (4) Since --→ BN = - --→ DN (because N is the midpoint of BD), by adding equations (3) and (4) we obtain that 2 --→ MN = --→ MB + --→ MD. Similarly, we obtain that 2 --→ NM = --→ NA + --→ NC. (5) The last two relations give that ( --→ MB + --→ MD ) 2 = ( --→ NA + --→ NC ) 2 (here by the square we mean
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