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 ?

Question

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  ?

crearti2d4 · Accepted Answer

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              .

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.

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