The intersection of 2 planes is constantly a line

If 2 planes intersect each other, the intersection will always be a line.

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The vector equation because that the heat of intersection is provided by

???r=r_0+tv???

where ???r_0??? is a allude on the line and ???v??? is the vector an outcome of the cross product that the normal vectors that the 2 planes.

The parametric equations because that the line of intersection are given by

???x=a???, ???y=b???, and also ???z=c???

where ???a???, ???b??? and also ???c??? space the coefficients from the vector equation ???r=aold i+bold j+cold k???.


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Finding the parametric equations that stand for the heat of intersection of two planes



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Example trouble of how to discover the heat where 2 planes intersect, in parametric for

Example

Find the parametric equations because that the heat of intersection the the planes.

???2x+y-z=3???

???x-y+z=3???

We need to discover the vector equation the the line of intersection. In stimulate to obtain it, we’ll require to an initial find ???v???, the cross product that the regular vectors of the provided planes.

The regular vectors because that the airplane are

For the aircraft ???2x+y-z=3???, the regular vector is ???alangle2,1,-1 angle???

For the aircraft ???x-y+z=3???, the typical vector is ???blangle1,-1,1 angle???

The cross product of the normal vectors is



We additionally need a point on the heat of intersection. To get it, we’ll usage the equations that the offered planes as a mechanism of linear equations. If we set ???z=0??? in both equations, we get

???2x+y-z=3???

???2x+y-0=3???

???2x+y=3???

and

???x-y+z=3???

???x-y+0=3???

???x-y=3???


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To uncover the line of intersection, first find a suggest on the line, and the overcome product of the typical vectors


Now we’ll add the equations together.

???(2x+x)+(y-y)=3+3???

???3x+0=6???

???x=2???

Plugging ???x=2??? ago into ???x-y=3???, we get

???2-y=3???

???-y=1???

???y=-1???

Putting these values together, the allude on the heat of intersection is

???(2,-1,0)???

???r_0=2old i-old j+0old k???

???r_0=langle 2,-1,0 angle???

Now we’ll plug ???v??? and also ???r_0??? into the vector equation.

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???r=r_0+tv???

???r=(2old i-old j+0old k)+t(0old i-3old j-3old k)???

???r=2old i-old j+0old k+0old it-3old jt-3old kt???

???r=2old i-old j-3old jt-3old kt???

???r=(2)old i+(-1-3t)old j+(-3t)old k???

With the vector equation because that the line of intersection in hand, us can discover the parametric equations because that the very same line. Equivalent up ???r=aold i+bold j+cold k??? with our vector equation ???r=(2)old i+(-1-3t)old j+(-3t)old k???, we deserve to say that

???a=2???

???b=-1-3t???

???c=-3t???

Therefore, the parametric equations because that the heat of intersection are

???x=2???

???y=-1-3t???

???z=-3t???

For integrals include exponential functions, shot using the power for the substitution.