Engineering Calculus Notes 100

Engineering Calculus Notes 100 - 88 CHAPTER 1. COORDINATES...

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Unformatted text preview: 88 CHAPTER 1. COORDINATES AND VECTORS Similarly, → a1 − = 2A (△OP1 Q1 ) ı → =− ı y1 z1 y2 z2 . Finally, noting that the direction from which the positive z -axis is → counterclockwise from the positive x-axis is −− , we have → a2 − = 2A (△OP2 Q2 ) → = −− x1 z1 x2 z2 . Adding these yields the desired formula. In each projection, we used the 2 × 2 determinant obtained by omitting the coordinate along whose axis we were projecting. The resulting formula can → → be summarized in terms of the array of coordinates of − and − v w x1 y1 z1 x2 y2 z2 by saying: the coefficient of the standard basis vector in a given coordinate direction is the 2 × 2 determinant obtained by eliminating the corresponding column from the above array, and multiplying by −1 for the second column. We can make this even more “visual” by defining 3 × 3 determinants. A 3 × 3 matrix 17 is an array consisting of three rows of three entries each, vertically aligned in three columns. It is sometimes convenient to label the entries of an abstract 3 × 3 matrix using a single letter with a double index: the entry in the ith row and j th column of a matrix A is denoted 18 aij , giving the general form for a 3 × 3 matrix a11 a12 a13 A = a21 a22 a23 . a31 a32 a33 We define the determinant of a 3 × 3 matrix as follows: for each entry a1j in the first row, its minor is the 2 × 2 matrix A1j obtained by deleting 17 Pronounced “3 by 3 matrix” Note that the row index precedes the column index: aji is in the j th row and ith column, a very different place in the matrix. 18 ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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