Practice Exam 1 - x & y + z = 2 and 3 x + y...

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Warning: The following is a practice exam. Do not expect the actual exam to necessarily consist of the same or similar problems. 1. (a) Find the area of the parallelogram with vertices P (1 ; 1 ; 2) , Q (6 ; 3 ; 6) , R (11 ; 1 ; 13) , and S (6 ; 3 ; 9) . (b) Find the angle at each vertex of this parallelogram. 2. True or False (Circle one.) Assume all vectors are three-dimensional. (a) True or False The angle between ~a and ~ b is obtuse ( 2 ) if and only if ~a ± ~ b < 0 . (b) True or False ~a is orthogonal to ~ b if and only if jj ~a ² ~ b jj = jj ~a jj jj ~ b jj : (c) True or False Vectors ~a , ~ b , and ~ c are coplanar if and only if the scalar triple product ~a ± ( ~ b ² ~ c ) = 0 . (d) True or False If ~a ± ~ b = ~a ± ~ c and ~a ² ~ b = ~a ² ~ c , then either ~a = ~ 0 or ~ b = ~ c: (e) True or False Two lines are parallel if and only if the cross product of their direction vectors is the zero vector. 3. Find parametric equations for the line of intersection of the planes
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Unformatted text preview: x &amp; y + z = 2 and 3 x + y &amp; z = 1 . 4. Find the equation of the plane which contains the point P (1 ; 2 ; 1) and the line with symmetric equations x &amp; 2 = y +2 2 , z = 4 . 5. Classify the surfaces de&amp;ned by the following equations as either a plane, cylinder, ellipsoid, elliptic paraboloid, hyperbolic paraboloid, hyperboloid of one sheet, hyperboloid of two sheets, or cone. (a) ( x + 1 &amp; z )( x + 1 + z ) + y 2 + z 2 = 4 . (b) x 2 &amp; 2 y + 8 z = y 2 + 4 z 2 . (c) x 2 = y 2 + 16 z 2 + 2 x . 6. Determine the length of the curve r ( t ) = h 2 t 3 ; 3 t 2 ; &amp; 3 t i , t a , in terms of a . 7. Find the tangential and normal components of acceleration of r ( t ) = h t; t cos t; t sin t i at the point ( ; &amp; ; 0) ....
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This note was uploaded on 09/08/2011 for the course MATH 21-259 taught by Professor Flaherty during the Spring '08 term at Carnegie Mellon.

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