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Calculus III Exam #1 Fall 2005 Name_____________________________ Show all your work on this paper. Solutions without correct supporting work will not be accepted. 1. Let u 4,1,5 and v 3,2,2 , find: (6 points each) a. u v b. the angle between u and v (approximate the nearest tenth of a degree) c. a vector orthogonal to u d. a unit vector in the direction of v e. the area of the parallelogram with adjacent sides u and v 2. Find a set of parametric equations for the line that passes through the point (1,2,3) and perpendicular to the plane given by 3( x 5) 2( y 1) ( z 4) 0 . (8 points) 3. An object is pulled 8 feet across a floor using a force of 75 pounds. Find the work done if the direction of the force is 30 degrees above the horizontal. (6 points) 3 3 3 3 , convert to , , 4. Given the rectangular coordinates of a point 4 4 2 (5 points each) a. cylindrical coordinates b. spherical coordinates 5. a. Write the equation of the plane containing the points (2,1,1), (0,4,1) and (-2,1,4). b. Sketch a graph of the plane in part (a). (4 points) (8 points) 6. Determine the maximum height and range of a projectile fired at a height of 3 feet above the ground with a initial velocity of 900 feet per second and at an angle of 45˚ above the horizontal. (10 points) 7. Find the tangential component of acceleration for the position vector given by r (t ) 3t i t j t 2 k . (8 points) 8. a. Find the unit tangent vector for r (t ) 2 cos t ,2 sin t , t at the point (2,0,0). (6 points) b. Find a set of parametric equations for the line tangent to the space curve at the point in part (a). (4 points) 9. Find the length of the space curve given by r (t ) 3t ,2 cos t ,2 sin t over the interval 0, . 2 (6 points)