Find the equation of a plane that goes through the intersection of the planes x 1 + 5 x 2 + x 3 − 5 = 0 and x 1 − 3 x 3 − 2 x 3 = 0 and contains the point (1,1,0)I calculated the cross product of the given plane to obtain the intersection and it gave me − 7 x 1 + 3 x 2 − 8 x 3 , but I'm stuck there.

Question

Find the equation of a plane that goes through the intersection of the planes       x    1    +  5      x    2    +      x    3    −  5  =  0 and       x    1    −  3      x    3    −  2      x    3    =  0 and contains the point (1,1,0)I calculated the cross product of the given plane to obtain the intersection and it gave me   −  7      x    1    +  3      x    2    −  8      x    3  , but I&#039;m stuck there.

Rebecca Benitez · Accepted Answer

Given  α  :      x    1    +  5      x    2    +      x    3    −  5  =  0  ,    β  :      x    1    −  3      x    2    −  2      x    3    =  0Any linear combination of the two planes passes through the intersection line                    (1)                    λ        (                  x          1                +        5                  x          2                +                  x          3                −        5        )        +        μ        (                  x          1                −        3                  x          2                −        2                  x          3                )        =        0            plug the coordinates of the point (1,1,0)  λ  (  1  +  5  +  0  −  5  )  +  μ  (  1  −  3  −  0  )  =  0  →  λ  =  2  μsubstitute in (1)  2  μ  (      x    1    +  5      x    2    +      x    3    −  5  )  +  μ  (      x    1    −  3      x    2    −  2      x    3    )  =  0  2  (      x    1    +  5      x    2    +      x    3    −  5  )  +  (      x    1    −  3      x    2    −  2      x    3    )  =  0  →  3      x    1    +  7      x    2    −  10  =  0

Siena Erickson · Accepted Answer

You need two vectors for the plane you want, the one you got is(−7,3,−8), so you need to find another one, choose a point P which lies both on       x    1    +  5      x    2    +      x    3    −  5  =  0 and       x    1    −  3      x    2    −  2      x    3    =  0 (I assume you had a type mistake), and the vector from P to (1,1,0) will be the second vector, do cross product then you will have the normal vector of the plane you want, (1,1,0) must be contained so you can get the constant from (1,1,0).

Find the equation of a plane that goes through the intersection of the planes x_1+5_x2+x_3−5=0 and x_1−3x_3−2x_3=0 and contains the point (1,1,0) .

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