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prelim1_spring2006solutions

prelim1_spring2006solutions - Math 192 Prelim 1 Solutions...

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Unformatted text preview: Math 192 Prelim 1 Solutions, Feb. 21, 2006 1. a) ~u ¡ ~v = 14 , | ~u | = √ 21 , | ~v | = 7 → θ = 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 Ī€ Ī€/ 2 tdt = 3 Ī€ 2 / 8. b) s ( t ) = R t Ī„dĪ„ = 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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