Find the intersection of a line formed by the intersection of two planes vec(r) vec(n)_1=p_1 and vec(r). vec(n)_2=p_2.

gobeurzb 2022-09-26 Answered
Find the intersection of a line formed by the intersection of two planes r . n 1 = p 1 and r . n 2 = p 2
I know that the line would be along ( n 1 × n 2 ). So i need a point on the line to get the equation. I assumed a point C such that O C is perpendicular to the line of intersection. I dont really know how to proceed from here. Do I have to use the equations c . n 1 = p 1 and c . n 2 = p 2 ?
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Answers (1)

crearti2d4
Answered 2022-09-27 Author has 9 answers
You know that the line is of the form t ( n 1 × n 2 ) + p for some point p. Now to find p, we can assume it is of the form a n 1 + b n 2 (since the line is perpendicular to n 1 and n 2 , so it must pass through the plane spanned by them somewhere). Then since p is in both planes, we have the equations p n 1 = a + b n 1 n 2 = p 1 , and p n 2 = a n 1 n 2 + b = p 2 . This gives us the system of equations
( 1 n 1 n 2 n 1 n 2 1 ) ( a b ) = ( p 1 p 2 ) .
The determinant of the matrix is 1 ( n 1 n 2 ) 2 , and since n 1 and n 2 are not parallel (since the planes intersect in a line, so they are not themselves parallel), this is positive. Hence we can invert the matrix to get
( a b ) = 1 1 ( n 1 n 2 ) 2 ( 1 n 1 n 2 n 1 n 2 1 ) ( p 1 p 2 ) ,
or letting n 1 n 2 = α
a = p 1 α p 2 1 α 2 ,
and
b = p 2 α p 1 1 α 2 .
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