## Two intersecting airplane always kind a line

If two planes intersect each other, the intersection will constantly be a line.

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The vector equation for the line of intersection is provided by

???r=r_0+tv???

where ???r_0??? is a point on the line and also ???v??? is the vector result of the overcome product that the common vectors of the 2 planes.

The symmetric equations for the heat of intersection are provided by

???fracx-a_1v_1=fracy-a_2v_2=fracz-a_3v_3???

where ???a(a_1,a_2,a_3)??? room the collaborates from a suggest on the heat of intersection and ???v_1???, ???v_2??? and also ???v_3??? come from the overcome product the the typical vectors come the given planes.

## Step-by-step instance of finding the intersection line together a symmetric equation

Example

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

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

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

We require to find the vector equation the the line of intersection. In stimulate to acquire it, we’ll need to first find ???v???, the cross product that the typical vectors the the provided planes.

The common vectors for the plane are

???alangle2,1,-1 angle??? because that the plane ???2x+y-z=3???

???blangle1,-1,1 angle??? for the plane???x-y+z=3???

The overcome product of the typical vectors is

We additionally need a point of top top the heat of intersection. To get it, we’ll usage the equations the the provided planes as a system of straight equations. If we set ???z=0??? in both equation, us get

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

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

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

and

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

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

???x-y=3???

Now we’ll include the equations together.

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???(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???

To uncover the symmetric equations, you’ll need the overcome product the the typical vectors the the two planes, and also a point on the line of intersection

Putting these worths together, the point on the line of intersection is

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

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

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

With the cross product that the regular vectors and the point on the line of intersection, we can plug into the formula because that the symmetric equations, and get

???x-2???, ???fracy-(-1)-3=fracz-0-3???

???x-2???, ???-fracy+13=-fracz3???

Remember, since the direction number because that ???x??? indigenous the cross product ???v=|a imes b|=langle0,-3,-3 angle??? is ???0???, we have to pull the symmetric equation for ???x??? far from the various other two and also keep that by chin so that we don’t have to divide by ???0???.

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find out mathKrista KingJanuary 21, 2019math, learn online, virtual course, virtual math, symmetric equations, heat of intersection, intersection of planes, aircraft intersection, partial derivatives