exam1a - Name Math5B-F08 Midterm I A Work in the space...

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Name Math5B-F08 Midterm I A Work in the space provided (you may use the back). No books, notes, or calculators. 5 problems on 5 pages, 95 points, 5 bonus points for clean and neat work. Always show work and/or reasoning. Circle your final answer! Problem 1 (25 pts: 11,6,8) (a) The points (0 , 0 , 0) , (1 , 0 , - 1), and (2 , 1 , 2), form the vertices of a space triangle. Find its area. (b) The vector F = 2 i - 3 j + k represents a force; what is its scalar component (=scalar projection) in the direction of 3 i + 4 k? (c) An origin vector in the first octant makes the same angle with each of the three coordinate axes. What is the angle? [Write it in the form cos - 1 ( α ).]
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Problem 2 (21 pts: 10,7,4) Consider the plane P : 2 x - 3 y - z = 14. (a) Find the point P 0 where a line passing through P 1 : (1 , 0 , 2) and perpendicular to P intersects P . (b) Find the equation of a plane P parallel to P and passing through P 1 . [Write the equation in the from ax + by + cz = d .] (c) Let P 2 be any point on P . The quantity ---→ P 1 P 2 · 2 , - 3 , - 1 | 2 , - 3 , - 1 | does not depend on the choice of P 2 .
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