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Sample exam for Math 251-510/511 Based solely on 11.1–11.3 c 2007, Gregory Berkolaiko Estimated time: 50 minutes. 1. (10 points) Given three points, A = (3, 0, 2), B = (4, 2, 0) and C = (5, 1, 4), determine if the triangle ABC is isosceles, right, both or neither. 2. (10 points) Find the scalar and vector projections of b = h4, 2, 0i onto a = h1, 1, 1i. 3. (5 points) Under what circumstances is compa b = compb a ? 4. (15 points) Find the distance from the point P (2, 0, 3) to the sphere given by the equation x2 + y 2 + z 2 − 2x + 4y − 2z + 5 = 0. (Hint: Find the center and the radius of the sphere to start with.) 5. (20 points) Find the two vectors that are √ • of length 2 and • perpendicular to a = (1, −1, 0) and • form π/3 angle with the vector b = (0, 1, 1). (Hint: cos(π/3) = 1/2.) 1