midterm1solutions

# midterm1solutions - MATH 20C — FIRST MIDTERM...

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Unformatted text preview: MATH 20C — FIRST MIDTERM SOLUTIONS (VERSION 1) Problem 1. Let v = h 2 , , 3 i and w = h 2 , 1 ,- 2 i . (a) (3 points) Find the cosine of the angle between v and w . The dot product is v · w = 2 · 2+0 · 1+3 · (- 2) = 4+0+(- 6) =- 2. The magnitudes are k v k = √ 2 2 + 0 2 + 3 2 = √ 13 and k w k = p 2 2 + 1 2 + (- 2) 2 = √ 9 = 3. Thus, the cosine of the angle is v · w k v kk w k =- 2 3 √ 13 . (b) (3 points) Decompose v = v k + v ⊥ into parts parallel and perpendicular to w . The parallel part is v k = ( v · w w · w ) w =- 2 9 h 2 , 1 ,- 2 i =- 4 9 ,- 2 9 , 4 9 . Thus, the perpendicular part is v ⊥ = v- v k = h 2 , , 3 i -- 4 9 ,- 2 9 , 4 9 = 22 9 , 2 9 , 23 9 . Problem 2. Let A = (1 ,- 2 , 0) , B = (3 , 2 , 2) , and C = (3 , 1 , 1) . (a) (3 points) What is the area of the triangle ABC ? Consider the vectors # — AB = h 3- 1 , 2- (- 2) , 2- i = h 2 , 4 , 2 i # — AC = h 3- 1 , 1- (- 2) , 1- i = h 2 , 3 , 1 i Their cross product is # — AB × # — AC = i j k 2 4 2 2 3 1 = (4- 6) i- (2- 4) j + (6- 8) k = h- 2 , 2 ,- 2 i ....
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midterm1solutions - MATH 20C — FIRST MIDTERM...

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