What is the intersection of planes called?

If a line and a plane intersect one another, the intersection will be a single point, or a line (if the line lies in the plane).

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To find the point of intersection, we’ll

1. substitute the values of ???x???, ???y??? and ???z??? from the equation of the line into the equation of the plane and solve for the parameter ???t???

2. take the value of ???t??? and plug it back into the equation of the line

This will give us the coordinates of the point of intersection.

 

 

The intersection of a line and a plane will either be a single point or a line

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Intersection of a plane and a line given by parametric equations

Example

Find the point where the line intersects the plane.

The line is given by ???x=-1+2t???, ???y=4-5t???, and ???z=1+t???

The plane is given by ???2x-3y+z=3???

Our first step is to plug the values for ???x???, ???y??? and ???z??? given by the equation of the line into the equation of the plane.

???2(-1+2t)-3(4-5t)+(1+t)=3???

???-2+4t-12+15t+1+t=3???

???20t=16???

???t=\frac{16}{20}???

???t=\frac45???

Now we’ll plug the value we found for ???t??? back into the equation of the line.

???x=-1+2\left(\frac45\right)???

???x=\frac35???

and

???y=4-5\left(\frac45\right)???

???y=0???

and

???z=1+\left(\frac45\right)???

???z=\frac95???

If a line and a plane intersect one another, the intersection will be a single point, or a line (if the line lies in the plane).

Putting these values together, we can say the point of intersection of the line and the plane is the coordinate point

???\left(\frac35,0,\frac95\right)???

If we want to double-check ourselves, we can plug this coordinate point back into the equation of the plane.

???2\left(\frac35\right)-3(0)+\left(\frac95\right)=3???

???\frac65+\frac95=3???

???\frac{15}{5}=3???

???3=3???

Since ???3=3??? is true, we know that the point we found is a true intersection point with the plane.

 

 

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Learn mathKrista KingOctober 22, 2020math, learn online, online course, online math, calculus 3, calculus iii, calc 3, calc iii, partial derivatives, line and plane, intersection, intersection of line and plane, intersecting line and plane

Two planes always intersect in a line as long as they are not parallel. Let the planes be specified in Hessian normal form, then the line of intersection must be perpendicular to both

and , which means it is parallel to

(1)

To uniquely specify the line, it is necessary to also find a particular point on it. This can be determined by finding a point that is simultaneously on both planes, i.e., a point

that satisfies

(2)

(3)

In general, this system is underdetermined, but a particular solution can be found by setting

(assuming the -component of is not 0; or another analogous condition otherwise) and solving. The equation of the line of intersection is then

(4)

(Gellert et al. 1989, p. 542). A general approach avoiding the special treatment needed above is to define

(5)

(6)

Then use a linear solving technique to find a particular solution

to , and the direction vector will be given by the null space of .

Let three planes be specified by a triple of points

where , 2, 3, denotes the plane number and denotes the th point of the th plane. The point of concurrence can be obtained straightforwardly (if laboriously) by simultaneously solving the three equations arising from the coplanarity of each of the planes with , i.e.,

(7)

for

, 2, 3 using Cramer's rule.

If the three planes are each specified by a point

and a unit normal vector , then the unique point of intersection is given by

(8)

where

is the determinant of the matrix formed by writing the vectors side-by-side. If two of the planes are parallel, then

(9)

and there is no intersection (Gellert et al. 1989, p. 542; Goldman 1990). This condition can be checked easily for planes in Hessian normal form.

A set of planes sharing a common line is called a sheaf of planes, while a set of planes sharing a common point is called a bundle of planes.

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