23 Sample Exam 1A - MATH 23 N AM E Sample First Exam was...

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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 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 tj - 2 t 2 k at the point P ( - 1 , 2 , - 2) .
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