Equation of line intersecting two planes

In summary, the cross product equation for the line L that forms the intersection of the planes A and B is (2,-1,2)⋅(x,y,z) = 1, where (x,y,z) represents any point on the line. By solving for specific values of x, y, and z, we can obtain parametric equations for the line. Another method is to use the equation n⃗ ⋅(r⃗ − r⃗ 0) = 0, where r⃗ 0 is a point that lies on the line.
  • #1
BigFlorida
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Homework Statement


[/B]
Find the cross product equation for the line L that forms the intersection of the planes:

A: 2x - y + 2z = 1 and B: x + y - 2z = 1.

Homework Equations



General equations for planes A and B:

(a, b, and r are vectors).

A: a dot r = 1

B: b dot r = 1

r = xi + yj + zk ; i, j, k here are the unit vectors.

r X (b X a) = b - a

alternatively, (b X a) X r = a - b

The Attempt at a Solution



I only know one way in which to solve for the equation of a line intersecting two planes and that is to use the coefficients of the variables of x, y, and z to define two normal vectors, cross those normal vectors to get a parallel line to the intersecting line, then set x, y, or z equal to zero and solve the equations for x and y to define a point. Then, having a vector in the direction of the line (the cross product of the norms) and a point on the line, put it in the form r(t) = (position vector of point) + t(cross product of the norms). I solved the equation for the line in this way, but my teacher wants it as a "cross product equation." I have no clue how to get a and b to stick into the equations for the planes listed above. The only thing I could think of was using the coefficients of x, y, and z; but that makes no sense because that would be dotting a vector with its normal vector and the product would be 0, not 1. Does anyone have any ideas? The parametric equation I got for the line was r(t) = < 2/3, 1/3, 0, 6, 3 >, but this is not in the r X (b X a) form he wants it in. Thanks in advance.
 
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Here is a simpler method: Any point on the intersection of the two planes must lie on both planes. And that means that if we designate the point (x, y, z), x, y, and z must satisfy both equations. Now, if we had three equations, we could, generally, solve for specific values of x, y, and z: three planes intersect in a single point.

With two equations, we can solve for two of the variables in terms of the third, using that third as parameter in parametric equations for a line.

Here, notice that one equation has "-y+ 2z" and the other has "y- 2z". Adding the equations, both y and z cancel, leaving a single equation in x only. Solve that for a specific value for x. The line is parallel to the yz-plane, x is that constant all along the line. Putting that value of x into the equations gives you one equation in y and z. Solve for y in terms of z and use z as parameter or solve for z in terms of y and use y as parameter.
 
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  • #3
BigFlorida said:
I have no clue how to get a and b to stick into the equations for the planes listed above. The only thing I could think of was using the coefficients of x, y, and z; but that makes no sense because that would be dotting a vector with its normal vector and the product would be 0, not 1.
Actually, your idea is right. Take the first plane, for example. You have ##2x-y+2z=1##, which can be written as ##(2,-1,2)\cdot(x,y,z) = 1##, which is the form you said you want.

An equation of a plane of the form ##\vec{n}\cdot \vec{r} = 0## corresponds to a plane that passes through the origin. For one that doesn't go through the origin, its equation can be expressed in the form ##\vec{n}\cdot (\vec{r} - \vec{r}_0) = 0##, where ##\vec{r}_0## is a point that lies in the plane.
 
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@HallsofIvy I did not think of doing that, but I will definitely give it a shot. Thank you!

@vela I never even considered the case of n⃗ ⋅(r⃗ − r⃗ 0) = 0, thank you so much. That has just helped me accomplish what I set out to do!
 

FAQ: Equation of line intersecting two planes

What is the equation of a line that intersects two planes?

The equation of a line intersecting two planes can be determined by finding the point of intersection between the two planes, and then using that point as the starting point for the line. The direction of the line can be determined by finding the cross product of the normal vectors of the two planes.

How do you find the point of intersection between two planes?

The point of intersection between two planes can be found by setting the equations of the two planes equal to each other and solving for the variables. The resulting values will give the coordinates of the point of intersection.

What is the significance of the normal vectors in finding the equation of the line?

The normal vectors of the two planes are crucial in determining the direction of the line that intersects them. The cross product of the two normal vectors gives the direction vector of the line, which is used in determining the slope of the line.

How many solutions can there be for the equation of a line intersecting two planes?

There can be three possible solutions for the equation of a line intersecting two planes: no solution, one solution, or infinitely many solutions. The number of solutions depends on the relationship between the two planes and their normal vectors.

Can the equation of the line intersecting two planes be written in different forms?

Yes, the equation of the line can be written in different forms such as parametric form, symmetric form, and vector form. Each form represents the same line but in a different way, and they can be converted to one another through simple algebraic manipulations.

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