Determine a so that the intersection line between the planes P 1 : 2 x + a y − z = 3 P 2 : x − 2 y + a z = 5are parallel to the plane P 3 : 2 x + y + z = 2

Question

Determine a so that the intersection line between the planes      P    1    :  2  x  +  a  y  −  z  =  3      P    2    :  x  −  2  y  +  a  z  =  5are parallel to the plane       P    3    :  2  x  +  y  +  z  =  2

nuvolor8 · Accepted Answer

Step 1You can directly solve for      |                            2                          a                          −          1                                      1                          −          2                          a                                      2                          1                          1                      |    =  0There are various justifications for this.First, as       P    1   and       P    2   have linear independent normals, there is a single line of intersection. Because that line does not pass through       P    3   , there is no solution (x,y,z) to the set of three equations, hence the determinant is 0.Secondly, the line is perpendicular to the normals of       P    1   and       P    2   so it&#039;s calculated via a cross product. The line is perpendicular to the third normal as well so their dot product is 0. Put it together we have the scalar triple product is 0.EDIT: One should verify that the line does not lie on the third plane, because both justifications also work if the planes meet at a single line (infinite solutions).

Quintin Stafford · Accepted Answer

Step 1Indirect approach: We first find the direction numbers of the line L resulting from intersection of two planes:  l  =      |                            a                          −          1                                      −          2                          a                      |    =      a    2    −  1  m  =      |                            −          1                          2                                      a                          1                      |    =  −  1  −  2  a  n  =      |                            2                          a                                      1                          −          2                      |    =  −  a  −  4The normal of plane are       N    p    :  (  A  =  2  ,  B  =  1  ,  C  =  1  ) and we must have   L  ⊥      N    p   ; the condition is that:  l  ×  A  +  m  ×  B  +  n  ×  C  =  0which gives this equation:  2      a    2    −  3  a  −  9  =  0which it&#039;s roots are   a  =  3 and   a  =  −      3    2  Using this method you can conclude the method mentioned in other answer.

Determine a so that the intersection line between the planes P 1 </msub> :

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