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l_notes04 - Lecture IV Analytic Geometry in E2 and E3 First...

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Lecture I Analytic Geometry in E 2 and E 3 First we review some basic facts of analytic geometry in E 2 . Let us consider a Cartesian coordinate system in E 2 . We denote by F [ x,y ] an algebraic formula in the variables x and y . Any equation of the form F 1 [ x,y ] = F 2 [ x,y ] can be reduced to an equation of the form F [ x,y ] = 0. We say that an equation F [ x,y ] = 0 determines a set S if S is the set of all points in E 2 whose coordinates satisfy the equation. In E 2 , an equation is called linear if has the form Ax + By + C = 0. where A,B, and C are real coefficients, with A 2 + B 2 > 0. Also, the following facts are true: (a) Every linear equation determines a unique straight line. (b) Every straight line is determined by some linear equation. Let vector N = A ˆ i + B ˆ j . Then vector N is normal to the line L determined by Ax + By + C = 0. The vector ˆ N = vector N | vector N | is a unit vector normal to the same line. Let O be the origin of the Cartesian system, let P be a point position vector vector R P .
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