Find Intersection of Plane & Line, Does Line Lie in Plane?

In summary, the person is trying to solve an equation for a line that is not in the plane. They need to use a 3D matrix rotator to coordinate rotate the entire problem so the line is vertical in rotated space, and then use the plane equation of the rotated line to solve for the z coordinate.
  • #1
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Homework Statement


Find the point(s) of intersection (if any) of the plane and the line. Also determine whether the line lies in the plane.

[tex]2x-2y+z=12, x-\frac{1}{2}=-y-\frac{3}{2}=\frac{x+1}{2}[/tex]


Homework Equations


[tex]2x-2y+z=12[/tex]

[tex]x-\frac{1}{2}=-y-\frac{3}{2}=\frac{x+1}{2}[/tex]


The Attempt at a Solution



I can seem to find the point of intersection just fine. What I'm having difficulty figuring out is how to determine whether the line lies in the plane or not.

I solved all the x, y, and z equations to t:

[tex]x=t+\frac{1}{2}[/tex]
[tex]y=-t-\frac{3}{2}[/tex]
[tex]z=2t-1[/tex]

I put the x, y, and z into the equation for the plane and solved for t:

[tex]2(t+\frac{1}{2})-2(-t-\frac{3}{2})+2t-1=12[/tex]

[tex]t=\frac{3}{2}[/tex]

Plug t=3/2 back into the equations for x, y, and z and I end up with the point of intersection at (2, -3, 2)

Now here is where I get stuck.

Shoot, while typing this I may have just came up with the solution but I'll ask and see if anyone can confirm.

Do I calculate the vector of the line and do a dot product to the normal of the plane. If the result is 0 then that means that the line lies on the plane?

Therefore
[tex]\vec{l}\bullet\vec{n}= <2,-2,1>\bullet<1,-1,2> = 2(1) + (-2)(-1) + 1(2) = 6[/tex]

So the line does not lie in the plane.

Is this assumption and my math correct? If not, where did I go wrong?
 
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  • #2


if the dot product of the normal to the plane & line is zero, you know the line is parallel to the plane, but you must also show it contains a point of the plane, to be contained by the plane

if the dot product is non-zero it cannot be contained in the plane
 
  • #3


i have the same problem but i don't understand how to tell if the line lies in the plane?
where does the <1,-1,2> come from?
 
  • #4


hey jcreed this post is 2 yrs old so you should probably open a new thread, you'll definitely get more answers that way as well

to show a line is contained is a plane you need to show
- there is a point on the line contained in the plane
- the line is parallel to the plane
This shows all points in the line are contained in the plane
 
  • #5


Algorithmic Geometry approach. This problem is straightforward if you know how to form generalized coordinate rotations (3D matrix rotators). The trick is to coordinate-rotate the entire problem so that the Line L stands perfectly vertical (Z-Axis-Aligned) in rotated space.

Inputs:
Line: defined by 3D points p1, p2
Plane: PL defined by equation: p • o == L, where: p is any point on the plane
o is the plane's orientation (normalized "normal")
L is the signed distance of plane from origin along direction o

Result: i is the intersection point


Steps to solve algorithmically:
0) bail if Line is parallel to plane
1) compute the run direction d of Line L = (p2 - p1)norm (normalized vector diff)
2) construct a Rotator with [ newXaxis, newYaxis, d ] where the software chooses the newXaxis, newYaxis dirVec pair arbitrarily
3) compute the invariantX'Y' coords of the rotated line L' by coordRotating p1 --> p1'
3) rotate the entire problem by the ZAlignRotator in 2)
a) computationally coordRotate PL by ZAlignRotator --> PL'
4) Use the plane equation of PL' : i' • PL'.o == PL'.L to solve for the z' coord of i'
5) coordUnrotate i' --> i

To determine if a Line lies within a Plane, simply generate 2 points on the line, and then test both to see if they lie in the plane (if they both satisfy the plane equation p • o == L )
 
  • #6


Hey pierre, welcome to PF

This is a pretty old post, so probably no need to re-open it.
 

1. What is the definition of the intersection of a plane and a line?

The intersection of a plane and a line is the point or set of points where the line and the plane meet. It is the solution to the simultaneous equations of the line and plane.

2. How do you find the intersection of a plane and a line?

To find the intersection of a plane and a line, you can use the method of substitution. This involves substituting the equations of the line into the equation of the plane and solving for the variables. The resulting values will give the coordinates of the intersection point.

3. Is it possible for a line to lie entirely in a plane?

Yes, it is possible for a line to lie entirely in a plane. This happens when the line's equation satisfies the equation of the plane. In this case, the line and the plane will have infinitely many intersection points along the entire length of the line.

4. Can a plane and a line intersect at more than one point?

No, a plane and a line can only intersect at one point. This is because a line is defined as a one-dimensional object, while a plane is defined as a two-dimensional object. Therefore, they can only intersect at one point in space.

5. What happens if the line and the plane are parallel?

If the line and the plane are parallel, then they will never intersect. This means that there is no solution to the simultaneous equations of the line and the plane. In other words, the line does not lie in the plane.

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