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Unformatted text preview: Math 192 Prelim 1 Solutions, Feb. 21, 2006 1. a) ~u ~v = 14 ,  ~u  = 21 ,  ~v  = 7 = cos 1 2 21 . b) ~w = 5( ~u + ~v  ~u + ~v  ) = 5 2 ~ j ~ k . c) i) False ii) True 2. a) The line of intersection is in the direction of ~n = < 2 , 1 , 3 > x < 1 , 1 , 2 > = < 5 , 7 , 1 > . Adding the equations for the planes gives us x + 5 z = 5. Letting x = 0 gives (0,1,1) as a point on the line. So the equations for the line are: x = 5t, y = 1  7t, z = 1 + t. ~n is normal to the plane and so the equation of the plane is 5( x 5) 7( y 6) + ( z + 3) = 0. b)We pick a point S(0,0,4) in one plane and apply the formula for the distance between a point and a plane. P(0,0,9) is in the other plane, so we have d = ~ PS < 2 , 3 , 1 >  < 2 , 3 , 1 >  = 5 / 14. 3. a) Arclength = R / 2 tdt = 3 2 / 8. b) s ( t ) = R t d = t 2 2 t = 2 s ~ r ( s ) = (cos 2 s + 2 s sin 2 s ) ~ i + (sin 2 s 2 s cos 2 s ) ~ j....
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This note was uploaded on 02/28/2008 for the course MATH 1920 taught by Professor Pantano during the Spring '06 term at Cornell University (Engineering School).
 Spring '06
 PANTANO
 Math, Calculus

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