MAT1341-L03-LinesandPlanes1

Case of r n lines let vector p and vectorv

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Case of R n Lines. Let vector P and vectorv negationslash = vector 0 be two vectors in R n . The line l passing through vector P and parallel to vectorv is given by the vector equation l : vector X ( t ) = vector P + tvectorv. This induces the parametric equations x 1 ( t ) = p 1 + tv 1 . . . x n ( t ) = p n + tv n . 4
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Planes. Let vectorv and vectorw be two non-collinear and non-zero vectors in R n . The plane through vector P parallel to vectorv and vectorw is given by the vector equation E : vector X ( s, t ) = vector P + tvectorv + svectorw. This induces the parametric equations x 1 ( t ) = p 1 + sv 1 + tw 1 . . . x n ( t ) = p n + sv n + tw n . Hyperplanes. Let vectorn negationslash = vector 0 and vector P be vectors in R n . The geometric object defined by the equation vectorn parenleftBig vectorx vector P parenrightBig = 0 is called a hyperplane . A hyperplane in R 2 is a line, and a hyperplane in R 3 is a plane. Cross Product Only for R 3 we can define a product of vectors. Let vectorv = ( v 1 v 2 v 3 ) T and vectorw = ( w 1 w 2 w 3 ) T be two vectors in R 3 . Their cross product is the vector defined as vectorv × vectorw = v 2 w 3 v 3 w 2 v 1 w 3 + v 3 w 1 v 1 w 2 v 2 w 1 . A direct computation shows the following results. Theorem 1. If vectorv and vectorw are non-zero vectors then vectorv × vectorw is orthogonal to vectorv and vectorw . Properties. Let vectoru , vectorv and vectorw be vectors in R 3 . Then vectorv × vector 0 = vector 0. vectorv × vectorv = vector 0. vectorv × vectorw = ( vectorv × vectorw ) . ( αvectorv ) × vectorw = α ( vectorv × vectorw ) = vectorv × ( αvectorw ). vectoru × ( vectorv + vectorw ) = ( vectoru × vectorv ) + ( vectoru × vectorw ). 5
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Intersection of Lines and Planes To find the relationship between two geometric objects (for example, are they parallel, orthogonal, etc) we will reduce the problem to a question about their direction vectors or their normal vectors. The problem of finding what is the intersection of two geometric objects will be solved by means of systems of linear equations. In general terms we have the objects do not intersect if the system has no solution; the intersection set consists of a single vector if the system has a unique solution; the intersection set is given by an infinite number of vectors if the system has multiple solutions. Example 1. Consider the lines l 1 : 5 x +3 y = 6 l 2 : 2 x = 18 (1) Is l 1 parallel to l 2 ? (2) Is l 1 orthogonal to l 2 ? (3) What is the intersection set? Normal vectors associated with l 1 and l 2 are parenleftbigg 5 3 parenrightbigg and parenleftbigg 2 0 parenrightbigg respectively.
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