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Unformatted text preview: LINES AND PLANES IN R n JOS ´ E MALAG ´ ON-L ´ OPEZ Case R 2 When we think of R 2 as a collection of points (not vectors), any line is determined by either any two points or a point and its slope. What happens when we think of R 2 as a collection of vectors? Let vector P be any vector whose end point is on the line l . In such case we say that the line l passes through vector P . Let vectorv be a vector parallel to l . In such case we say that vectorv is a direction vector of l . Note. A direction vector plays the role of the slope in this new setting. Remark. Notice that ALL the vectors which the line l passes through determine the line l . Let l , vector P and vectorv be as above. Let vectorw be such that l passes through it. W P V W-P Notice that vectorw − vector P is parallel to vectorv , so there exists a unique α negationslash = 0 such that αvectorv = vectorw − vector P. We conclude that (1) vectorw = vector P + αvectorv for some scalar α . 1 Summarizing, given vectors vectorv and vector P so that a line l passes through vector P and vectorv is parallel to l , we have that for any vector vectorw for which l passes through there is a unique number α so that the equation (1) holds. Vector Equation of a Line. Notice that as α runs over all the (real) numbers we can talk about the function (2) vector X ( t ) = vector P + tvectorv, where t is an independent variable. The function above determines the line l by giving us all the vectors for which l passes through. The function (2) is called the vector equation of l . Remark. The function vector X ( t ) = tvectorv determines a line through the origin. So, if vector P negationslash = vector 0 then the function vector X ( t ) = vector P + tvectorv is the “translation” of the line vector X ( t ) = tvectorv along the vector vector P . Remark. If y = mx + b determines a line l , the direction vector of l is given by vectorv = (1 , m ). Note. The function (2) is an example of a parametric representa- tion of a geometrical object. The independent variable is called the parameter . Parametric Equation of a Line. Given a line l : vector X ( t ) = vector P + tvectorv, with vector P = ( p 1 p 2 ) T and vectorv = ( v 1 v 2 ) T , the components of vector X ( t ) are given by (3) x 1 ( t ) = p 1 + t v 1 x 2 ( t ) = p 2 + t v 2 . The equations above are called the parametric equations of l . Normal Equation of a Line. We would like to represent a line by means of dot product. We say that a vector vectorn is normal to a line l if vectorn is orthogonal to any direction vector of l ....
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