# How To Find The Intersection Of Two Lines In Space If At All Possible.

• Baumer8993
In summary, to prove whether or not two lines in three space intersect, you can use a different parameter for each line and set the equations for x, y, and z equal to each other. This will give you two equations with two unknowns, which can be solved to determine if there is a common point of intersection. However, in general, two lines in three space do not intersect.
Baumer8993

## Homework Statement

I am given the equation of two lines that are in three space. They are in the form of (X=, Y=, Z= ). The questions wants me to prove whether or not the lines intersect.

## Homework Equations

The equations of the lines. It gives me just the points in the equation, but it is assumed I can get the equation of the line easily.

## The Attempt at a Solution

I guess I need to find a common point some how? I know the intersections may not be at the same time.

Baumer8993 said:

## Homework Statement

I am given the equation of two lines that are in three space. They are in the form of (X=, Y=, Z= ). The questions wants me to prove whether or not the lines intersect.

## Homework Equations

The equations of the lines. It gives me just the points in the equation, but it is assumed I can get the equation of the line easily.

## The Attempt at a Solution

I guess I need to find a common point some how? I know the intersections may not be at the same time.
Use a different parameter for each line. Instead of using t as the parameter for both lines, use, say, s for one line, and t for the other. At any point of intersection both x values have to be equal, both y values, and both z values.

Note that this will give you three equations (for x, y, and z) in the two unknowns s and t. You can use any two equations to solve for s and t and then you will have to check to see if they satisfy the third. That is why the "general" situation is that two lines in three space do NOT intersect.

## 1. How do I determine if two lines in space intersect?

To determine if two lines in space intersect, you can set up a system of equations that represent the two lines and solve for the point of intersection using algebraic methods. If the equations have a unique solution, then the lines intersect at a single point. If the equations have no solution, then the lines are parallel and do not intersect. If the equations have infinite solutions, then the lines lie on top of each other and intersect at every point along their length.

## 2. Is there a specific method for finding the intersection of two lines in space?

Yes, there are various methods for finding the intersection of two lines in space. These include using vector equations, parametric equations, or Cartesian equations. Each method has its advantages and disadvantages, so it is important to choose the method that is most suitable for the given problem.

## 3. Can I find the intersection of two lines in space without using equations?

No, it is not possible to find the intersection of two lines in space without using equations. The equations represent the relationships between the coordinates of the points on the lines and are necessary to determine the point of intersection.

## 4. What happens if the two lines in space are skew (do not lie in the same plane)?

If the two lines in space are skew, they will not intersect at a single point. Instead, they will have a closest point of approach, where the distance between the lines is minimized. This point can be found using vector projections or parametric equations.

## 5. Can I find the intersection of more than two lines in space?

Yes, it is possible to find the intersection of more than two lines in space. This can be done by solving a system of equations that represent all the lines and finding the common point of intersection. However, as the number of lines increases, the calculations become more complex and may require the use of computational methods.

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