mid1sol - Problem 1. Consider the points P (1 , 1 ,- 2), Q...

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Unformatted text preview: Problem 1. Consider the points P (1 , 1 ,- 2), Q (2 , , 1) and R (1 ,- 1 , 0). (i) Find the area of the triangle PQR . We calculate ~ PQ = (1 ,- 1 , 3) , ~ RQ = (1 , 1 , 1) . The area of the parallelogram spanned by ~ PQ and ~ QR is the magnitude of the cross product (2 ,- 1 , 3) (1 , 1 , 1) = (- 4 , 2 , 2) . This vector has magnitude 2 6, so the triangle PQR has area 6. (ii) Find the equation of the plane through P , Q and R . The plane through P, Q and R has as normal vector the cross product. The entries of the cross product are used as coefficients for the plane. We obtain the equation- 4 x + 2 y + 2 z =- 6 - 2 x + y + z =- 3 using the point P (or Q or R ) to find the right hand side. 1 Problem 2. (i) Does there exist a constant such that the function f ( x,y,z ) = ( x 4 y x 2 + y 2 + z 2 if ( x,y,z ) 6 = (0 , , 0) a if ( x,y,z ) = (0 , , 0) is continuous? We must have a = lim ( x,y,z ) (0 , , 0) x 4 y x 2 + y 2 + z 2 . Note that x 4 y x 2 + y 2 + z 2 = x 2 x 2 + y 2 + z 2 | x 2 y | | x 2 y | hence a = 0....
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mid1sol - Problem 1. Consider the points P (1 , 1 ,- 2), Q...

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