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hmwk6 - product(b Define a vector ~ b that is given by the...

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Department of Chemical Engineering University of California, Santa Barbara Eng 5A Fall 2001 Instructor: David Pine Homework #6 Homework due: Wednesday, 6 November 2001 1. Consider the following matrix: M = a b c 0 d e 0 0 f (a) Form the partitioned matrix [ M | I ] and row-reduce this to find the inverse M - 1 of M . Then verify that you have found the inverse by doing the multiplication M - 1 M . For this part use Mathematica only to multiply and/or add rows together. (b) Find the inverse M - 1 by defining the matrix A = [ M | I ] and then row reducing A using the RowReduce function of Mathematica . (c) Find the inverse M - 1 using the Inverse function of Mathematica . Verify that you obtain the same result for M - 1 using all three methods (d) Fine the determinant of M and M - 1 and verify that their product is unity. 2. The three points (1 , 1 , 2), (2 , 0 , 0), (0 , - 1 , 1) define a plane. Use vectors to find an ( x, y, z ) equation for the plane that contains these three points. Employ the following method using Mathematica . (a) Find a vector ~n perpendicular to the plane containing three points using the cross
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Unformatted text preview: product. (b) Define a vector ~ b that is given by the difference between an arbitrary point ( x, y, z ) and one of the three points that define the plane. Note that ( x, y, z ) will be in the plane defined by our three points if the vector ~ b is perpendicular to ~n . Use the Solve command of Mathematica to find the condition (equation for x , y , and z ) that satisfies the equation ~ b · ~n = 0. Verify that the three points satisfy the equation you find. (c) Use your solution to find a parametric expression for the plane in R 3 that contains the three points (1 , 1 , 2), (2 , , 0), (0 ,-1 , 1). That is, find the column matrix (vector) ~x such that ~x = ~ P + t 1 ~ b 1 + t 2 ~ b 2 , where ~x = ( x, y, z ), ~ P is a particular column matrix, and t 1 and t 2 are numbers (scalars), and where ( x, y, z ) are points in the plane defined by the three points (1 , 1 , 2), (2 , , 0), (0 ,-1 , 1)....
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