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Elements of Analytical Geometry and of the Differential and Integral Calculus

Elements of Analytical Geometry and of the Differential and Integral Calculus


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About the Book

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1860 Excerpt: ... If we designate the co-ordinates of a particular point, by x', y', z', the equations of the line passing through this point, will be, x-x' = a(z-z'). (1.) y-y' = b(z-z'). (2.) The equations of a second line passing through the same point, of which the co-ordinates are x', y', z are of the form, x--x' = a' (z--s').... (3.) y-y' = 'b'(z-z').... (4.) But the two lines will be at right angles to each other, if their equations fulfill the condition (Art. 23--1).. 1 + aa' + bb' = 0 (5.) If we now attribute to a' and b', all possible values that will satisfy this equation, we shall have all the perpendiculars which can be drawn to the given line, through the point whose co-ordinates are x', y', z'. These perpendiculars determine the plane.. Legendie, Bk. VI. Dof. 1. It is necessary, however, to find the equation of the plane in terms of the' co-ordinates of its different points. We find from Equations (3 ) and (4), z--z' Substituting these values in Equation (5), 1 + aa' + bb' = 0, and reducing, we find, z--z' + a(x--x') + by--y')--0; but, since a, b, z', x', y', are. known quantities, we may denote the constant part of the equation by a single letter, by making, --z'--ax'--by' =--c. hence, the equation of the plane becomes, z + ax + by--c = 0. 1. Since the equation of a plane contains three vari - ables, we may assign values, at pleasure, to two of them, and the equation will then make known the value of the third. For example, if we assign known values, denoted by x' and y', to x and y, the equation of the plane will give, z = c--ax'--by', and hence, the co-ordinate z becomes known. Traces of planes. 26. The lines in which a plane intersects the co-ordi nate planes, are called the traces of the plane. These. traces are found by combining the equation of the...


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Product Details
  • ISBN-13: 9781150906145
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 60
  • Spine Width: 3 mm
  • Width: 189 mm
  • ISBN-10: 1150906146
  • Publisher Date: 01 May 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 127 gr


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