Practice Exam 1 - P =(1-1(b[3 pts Find the point at which...

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MATH 241: CALCULUS III Department of Mathematics, UMCP Fall 2007 Practice Exam 1 Handed out : Friday, 09/28/07 WORK ON ALL PROBLEMS. Justify your answers. Cross out what is not meant to be part of your final answer. 1. Consider the vectors a = i + λ j + 3 k , b = i - j , c = i + 2 j + 4 k , where λ is a real number. (a)[4 pts] Find λ so that a and b are perpendicular. (b)[6 pts] For the value of λ found in part (a), resolve c into vectors parallel to a and b . Useful formula: You can use the formula for the projection of c onto a : pr a c = a · c || a || 2 ! a . 2. Consider the following vectors: a = 3 i + j - 5 k , b = i - k . (a)[7 pts] Find the equation of the plane that contains a and b and goes through the point
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Unformatted text preview: P = (1 , ,-1). (b)[3 pts] Find the point at which this plane intersects the line x = 3 t , y = 1 + 2 t , z =-1 + t . 3. (10 pts) Let C be the curve parametrized by the vector r ( t ) = ln t i + t 2 2 j + t √ 2 k , 1 ≤ t ≤ 2 . Find the arc length function s ( t ) for 1 ≤ t ≤ 2. 4. Consider the planes λx + 2 y + 3 z = 1 , x + y + z = 1 . (a)[4 pts] Find λ so that these planes are perpendicular. (b)[6 pts] For the value of λ found in part (a), ±nd an equation for the intersection l of the given planes....
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