samplx1s10 - L that goes through the point ± r(0 5(15...

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MATH 23 Sample First Exam was: February, 2003 NAME : Section (Last, First) 1. (15 points ) If ± U, ± V and ± W are the vectors ± U = ± i - 3 ± j + ± k, ± V = 2 ± i - ± j + 3 ± k and ± W = - ± i + 2 ± j - ± k, where ± i, ± j and ± k are the unit vectors in the direction of the coordinate axes, find the following. (a) the dot product of ± U and ± V (b) the scalar projection of ± V onto ± W (= component of ± V in the direction of ± W ) (c) the vector projection of ± V onto ± W 2. (10 points ) Find the center and radius of the sphere with equation x 2 - 6 x + y 2 + 2 y + z 2 = 3 . 3. (15 points ) (a) Find a vector perpendicular to the plane through (3 , 0 , - 1) , (2 , 1 , - 5) and (1 , 2 , - 4) . (b) give an equation for the plane in part (a). 4. Let L be the line given by the vector equation ± r ( t ) = < - 2 , 4 , 1 > + t < 3 , 2 , 4 > . (a) (5 points ) Find the point P that is on both the line L and the xz -plane (the one with equation y = 0). (b) (5 points ) Find an equation of the plane perpendicular to the line
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Unformatted text preview: L that goes through the point ± r (0) . 5. (15 points ) Find an equation for the tangent line to the curve-→ r ( t ) = t 3 ± i-2 t ± j-2 t 2 ± k at the point P (-1 , 2 ,-2) . 6. (15 points ) (a) If-→ r ( t ) = < t, 3 cos t, 3 sin t >, find the unit tangent vector ± T ( t ). (b) Find ± T and ± N at the point (0 , 3 , 0) . 7. (20 points ) The position function for the motion of a particle is given by-→ r ( t ) = ( 2 3 t 3 ) ± i + 2 t ± j + t 2 ± k = < 2 3 t 3 , 2 t, t 2 > . (a) Find the acceleration vector. (b) Find the tangential component a T of acceleration. (c) What is the normal component when t = 0? (= a N (0)) (d) How fast is the particle speeding up at time t = 1?...
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This note was uploaded on 03/26/2010 for the course MATH 23 taught by Professor Yukich during the Spring '06 term at Lehigh University .

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