Find the intersection of 2 x 1 + x 2 + x 3 − 3 = 0, 2 x 1 + x 2 + 4 x 3 − 6 = 0, 2 x 1 + x 2 − 2 = 0So I calculated the determinant of the three planes and it's equal to zero, then I calculated the cross product between the two first planes and it gave me 3 x 1 − 6 x 2 The thorem in the course book says that if the determinant of the three planes is equal to zero, and cross product it's not zero, then there's two posibilities if r 4 − u p 4 − v q 4 = 0 then the three planes intersect in a line l and l = { ( x 1 , x 2 , x 3 ) : S X = u p × q } and S is a point in the line.What is r 4 , u p 4 and v q 4 ?

Question

Find the intersection of  2      x    1    +      x    2    +      x    3    −  3  =  0,   2      x    1    +      x    2    +  4      x    3    −  6  =  0,   2      x    1    +      x    2    −  2  =  0So I calculated the determinant of the three planes and it&#039;s equal to zero, then I calculated the cross product between the two first planes and it gave me   3      x    1    −  6      x    2  The thorem in the course book says that if the determinant of the three planes is equal to zero, and cross product it&#039;s not zero, then there&#039;s two posibilities if       r    4    −  u      p    4    −  v      q    4    =  0 then the three planes intersect in a line l and   l  =  {  (      x    1    ,      x    2    ,      x    3    )  :  S  X  =  u  p  ×  q  } and S is a point in the line.What is       r    4  ,   u      p    4   and   v      q    4  ?

Adelyn Mercado · Accepted Answer

These three planes are pairwise non-identical and non-parallel so they will meet in a point, or a line or they will form an open prism.If you write it like Ax=B and if   det      |    A      |    =  0, the planes will either meet in a line or the will form an open prism. If you find the edges of pairwise planes by finding cross products of their normals                     n        →              1    ,                    n        →              2    ,                    n        →              3  . If edges are identical/coincident the planes will meet in a line. If edges are non coplanar planes will form a prism.But more simply here if you put (3) in (1) and (2), you get       x    3    =  1 and       x    3    =  2 contradiction (an inconsistency). So these three Eqs. are inconsistent and hence they form a prism. In a prism two plane meet leaving out one, so no solution (inconsistency).

Find the intersection of 2x_1+x_2+x_3−3=0, 2x_1+x_2+4x_3−6=0, 2x_1+x_2−2=0.

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