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# Equation of a line intersepting two planes

Posted 11 years ago
 HiI know equations of two planes and I want to find the line of interseption of those. But I do not know any point on that intercepting line. What I leart so far is that, by taking the cross product of the perpendicular vectors of the two planes, I can find the vector of the line. But without knowing a point on the line I can't find the line equation. Appreciate your help to solve this problem. Chin
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Posted 11 years ago
 The general method is as follows: Each plane (2D plane in 3D euclidean space) has a normal vector. The intersection of the two planes is a line and this line is normal to both of the normal vectors by definition. So it is the cross product of the two normals.
Posted 11 years ago
 @ChinthakaRatnaweera are you asking a purely math question? This is the site dedicated to posts ONLY about Wolfram Technologies. We do not address general math questions.
Posted 11 years ago
 Aculally I am looking for a general way of solving this. How to find the range of values that satisfy t (parameter)?This link explains a general method. http://mathworld.wolfram.com/Plane-PlaneIntersection.htmlBut I don't really follow Gellert et al's general approach. Great someone can explain this more.Thanks
Posted 11 years ago
 No need for a point. Use parameteric representation ClearAll[x, y, z, t]; p1 = x + 5 y + 2 z + 3 == 0; (*equation of first plane*) p2 = 2 x + 3 y + 6 z + 4 == 0; (*equation of second plane*) eq1 = Simplify[Inner[Subtract, Thread[2 *p1, Equal], p2, Equal]]; (*eliminate x*) zt = t; (*let z=t*) yt = y /. First@Solve[eq1 /. z -> zt, y];  (*find y in terms of t*) xt = x /. First@Solve[p1 /. {z -> t, y -> yt}, x]; (*find x in terms of t*)  Show[ Plot3D[z /. First@Solve[p1, z], {x, -2, 2}, {y, -3, 3},PlotStyle -> Green, Mesh -> None], Plot3D[z /. First@Solve[p2, z], {x, -2, 2}, {y, -3, 3},PlotStyle -> LightBlue, Mesh -> None], ParametricPlot3D[{xt, yt, zt}, {t, -3, 3}, PlotStyle -> {Thickness[.03], Red}], Axes -> None, Boxed -> False ]