MAT1341-L03-LinesandPlanes1

2 if l passes through a vector vectorw we have that

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If l passes through a vector vectorw we have that vector P vectorw is a vector direction of l (see figure above). This implies that vectorn ( vectorw vector P ) = 0. The vectors vectorw satisfying this identity determine the line l . The equation vectorn parenleftBig vectorx vector P parenrightBig = 0 is called the normal equation of the line l . General Equation of a Line. If we have that vectorn = ( A B ) T and vector P = ( p 1 p 2 ) T , and we set vectorx = ( x 1 x 2 ) T , we conclude 0 = parenleftbigg A B parenrightbigg parenleftbiggparenleftbigg x 1 x 2 parenrightbigg parenleftbigg p 1 p 2 parenrightbiggparenrightbigg = parenleftbigg A B parenrightbigg parenleftbigg x 1 p 1 x 2 p 2 parenrightbigg = Ax 1 + Bx 2 Ap 1 Bp 2 . If we set C = Ap 1 Bp 2 , we obtain the general equation of the line l : Ax 1 + Bx 2 + C = 0 . Case of R 3 Lines. In R 3 a line is still determined by two points, so we can proceed as we did for the equations (1) and (2). Given two vectors vector P and vectorv in R 3 , the line l passing through vector P and parallel to vectorv can be defined as follows: (Vector Equation) l : vector X ( t ) = vector P + tvectorv . (Parametric Equations) If vector P = ( p 1 p 2 p 3 ) T and vectorv = ( v 1 v 2 v 3 ) T , we have that l is given by x 1 ( t ) = p 1 + tv 1 x 2 ( t ) = p 2 + tv 2 x 3 ( t ) = p 3 + tv 3 Planes. However, in R 3 is not longer true that a line is determined by a vector normal to it (we have 3 different directions!). Let vectorv and vectorw be two vectors non-collinear in R 3 . Let s and t denote two parameters. The vector equation E : vector X ( s, t ) = svectorv + tvectorw defines a plane passing through the origin. 3
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Given a vector vector P , the equation (4) E : vector X ( s, t ) = vector P + svectorv + tvectorw defines the plane passing through vector P and parallel to the plane svectorv + tvectorw . If vector P = ( p 1 p 2 p 3 ) T , vectorv = ( v 1 v 2 v 3 ) T and vectorw = ( w 1 w 2 w 3 ) T , from the equation (4) we obtain the parametric equations x 1 ( t ) = p 1 + sv 1 + tw 1 x 2 ( t ) = p 2 + sv 2 + tw 2 x 3 ( t ) = p 3 + sv 3 + tw 3 of the plane E . We say that a vector vectorn in R 3 is normal to a plane E as given by the vector equation (4) if vectorn ( svectorv + tvectorw ) = 0. In particular, vectorn vectorv = 0 and vectorn vectorw = 0. Proceeding as for the case of R 2 , we get that the equation E : vectorn parenleftBig vector X vector P parenrightBig = 0 defines a plane in R 3 , with vectorn normal to it. In this case, the general equation of E is of the form Ax 1 + Bx 2 + Cx 3 = D. Note. Given a plane E : Ax 1 + Bx 2 + Cx 3 = D , the vector ( A B C ) T is normal to E .
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