5Bmidterm-2007-solution - Midterm Exam Math 5B November...

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Midterm Exam Math 5B November 2007 Your name: Your perm. #: Your signature: 1. 2. Subtotal: 3. 4. Subtotal: Total (100): When presenting your solutions, be sure to write down all the steps clearly and cleanly! If you don’t show work, and only write down your final result for a problem, you’ll receive no credit for that problem! 1
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1. (25 points) Let P be the plane which contains the points A = (1 , 2 , 1) , B = (2 , 1 , 2) and C = (2 , 3 , 4). (1) Find a normal vector to P . (2) Find an equation for P . (3) Find the area of the triangle with A, B and C as the vertices. (Hint: consider a suitable parallelogram.) Solution (1) We have -→ AB = (2 - 1 , 1 - 2 , 2 - 1) = (1 , - 1 , 1) , (1) -→ AC = (2 - 1 , 3 - 2 , 4 - 1) = (1 , 1 , 3) . (2) Then a desired normal vector is given by -→ AB × -→ AC . There holds -→ AB × -→ AC = i j k 1 - 1 1 1 1 3 = - 4 i - 2 j + 2 k . (3) 2) We have the normal vector given above. We take A as the reference point. Then the desired equation is given by ( x - 1 , y - 2 , z - 1) · ( - 4 , - 2 , 2) = 0 , (4) which simplifies to 2 x + y - z = 3 . (5) 3) The desired area is given by half of the area of the parallelogram formed by -→ AB and
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