:PROPERTIES: :ID: e4a53d0e-6b4e-43db-8752-e5e17c8b9583 :mtime: 20231010013724 :ctime: 20231010013545 :END: #+title: intersection of 2 planes #+filetags: :public:project: * Problem Definition The "interesection of two planes" problem refers to a problem where you are given two planes, and asked to find the line which is their intersection. * Example Problem Find the line of intersection of the two planes with normals parallel to the vectors $(1,0,2)$ and $(-1,1,1)$ respectively, which both pass through the point $(1,1,0)$. Which angles does this line matke with the coordinate axes?? ** Solution The first plane has [[id:53d108df-7937-4e8f-9f07-686b681cf0e8][equation]] $(x - 1 , y - 1 , z) \cdot (1 , 0 , 2 ) = 0 = x - 1 + 2z = 0$ The second plane has [[id:53d108df-7937-4e8f-9f07-686b681cf0e8][equation]] $(x - 1 , y - 1 , z) \cdot (-1,1,1) = -x + 1 + y - 1 + z = -x + y + z = 0$ We can substitute these two equations together to get $z = \frac{1-x}{2} = \frac{1-y}{3}$. We can recognize that this is the [[id:dded15e1-eb8c-4576-8c9f-818ea7b80ac1][component equation of a line]]. This means that the line goes through the points $(1,1,0)$ and $(-2,-3,1)$. Using these points, we can find the angles with each of the axes. * Example: Visibility over a wall Can an observer $A$ at $(4,5,1)$ see object at the point $B$ at $(2,3, \frac{3}{2})$ when there is an intervening wall with the top given by line $\vec{r} = (1,2,1) + \lambda(1,-1,-1)$? ** Solution TODO