# Intersection of Lines & Planes: Find Point?

• EvaBugs
In summary, this person is not sure how to find the intersection of two lines. They suggest using the parametric form of the lines and solving for the dist. If the distance is zero, then the parameters t1,t2 will give you the same point.
EvaBugs
To find the point of intersection of two lines, do I use the same method as in finding the intersection of a line and a plane?

what method do you use to find the intersection of a line and a plane?

The line intersects the xy-plane when z=0

well, here's what i do to find the point of intersection of two lines. i just set them equal to each other.

ooh a graphics question...sorry i can't remember the solution off the top of my head its in my graphics book but assuming you no that the 2 lines intersect then the solution
evolves using the parametrics

aight i go get the book...

This ist he distance between 2 lines: obviously if there is an intersection
then d=0 but the solution points still hold true to what your looking for

|t1| = |V1.V1 ,-V1.V2|^-1 * |(P2-P1).V1|
|t2|... |V1.V2 ,-v2.v2|...|(P2-P1).V2|
hope this doesn't look ugly

t1,t2 will give you the parametric solution plug into one and you get your point.

neurocomp2003 said:
ooh a graphics question...sorry i can't remember the solution off the top of my head its in my graphics book but assuming you no that the 2 lines intersect then the solution
evolves using the parametrics

aight i go get the book...

This ist he distance between 2 lines: obviously if there is an intersection
then d=0 but the solution points still hold true to what your looking for

|t1| = |V1.V1 ,-V1.V2|^-1 * |(P2-P1).V1|
|t2|... |V1.V2 ,-v2.v2|...|(P2-P1).V2|
hope this doesn't look ugly

t1,t2 will give you the parametric solution plug into one and you get your point.

I'm sorry, but I don't really understand the forumla. Is that the distance formula?

Should I find the line that is the cross-product of the normal vectors?

its a matrix form(2 equation Linear system)formula for distance...and if the distance is zero than the parameters t1,t2 find the same point...sorry but i don't know how to make the spaces stick.

The formula(from a math game book but found in any text)
takes the parametric form of lines and then solves the dist...using minimization and PDEs it results in the 2 equation linear system above.

Last edited:
EvaBugs said:
Should I find the line that is the cross-product of the normal vectors?

What form are you equations in? Lines in three dimensions are usually written by specifying a point on the line as a vector, plus a multiple of a vector parallel to the line. The multiplying constant is called a parameter, often designated by t

$$\overrightarrow r = \overrightarrow r _0 + t\overrightarrow r _\parallel$$

Another way of specifying a line is to eliminate the parameter and write the multiple equalities

$$\frac{{x - x_0 }}{a} = \frac{{y - y_0 }}{b} = \frac{{z - z_0 }}{c}$$

Whichever way your lines are written, to find their intersection you need to have the lines passing through the same point in space. For the first form of the equation, that means

$$\overrightarrow r_1 = \overrightarrow r _{0,1} + t_1\overrightarrow r _{\parallel,1} = \overrightarrow r_2 = \overrightarrow r _{0,2} + t_2\overrightarrow r _{\parallel,2}$$

I think this is probably easier to deal with than the second form. It is a vector equation, which is really three equations. The only way two vectors can be equal is if their individual components are each equal.

Last edited:
Another way to look at this situation:

If your equations are in vector form, consider the vector

$$\overrightarrow r_2 - \overrightarrow r_1$$

The two lines go through the points that are defined by these vectors. For the lines to intersect, they must be coplaner with this difference vector. Do you know how to find out if three vectors lie in the same plane?

## 1. How do you find the intersection point of two lines?

To find the intersection point of two lines, you can use the method of substitution or elimination. First, set the equations of the two lines equal to each other and solve for one variable. Then, substitute this value into the other equation to find the corresponding value for the other variable. This will give you the coordinates of the intersection point.

## 2. What is the equation for the intersection of a line and a plane?

The equation for the intersection of a line and a plane is the solution to the system of equations formed by the equation of the line and the equation of the plane. This can be found by substituting the values of the line's variables into the plane's equation and solving for the remaining variable.

## 3. How do you determine if a line intersects with a plane?

A line will intersect with a plane if their equations are consistent, meaning they have at least one solution when solved simultaneously. This can be determined by checking if the coefficients of the variables in both equations are proportional or not. If they are not proportional, then the line and plane will intersect at a single point.

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

No, a line and a plane can only intersect at one point. This is because a line is one-dimensional and can only have one point of intersection with a two-dimensional plane.

## 5. How can you find the intersection point of two planes?

To find the intersection point of two planes, you can use the method of substitution or elimination. First, set the equations of the two planes equal to each other and solve for one variable. Then, substitute this value into the other equation to find the corresponding values for the remaining variables. This will give you the coordinates of the intersection point.

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