Finding Equation of line of intersecting planes

In summary, the equation of a line of intersecting planes can be found by determining the point of intersection between two planes and using it as the starting point for the equation. The direction vector of the line is found by taking the cross product of the normal vectors of the two planes. The significance of the direction vector is that it represents the direction in which the line is moving and helps to define its orientation. The equation can be simplified by using the unit vector in the direction of the line. Two planes must intersect in order for a line to be formed.
  • #1
Larrytsai
228
0
Hey guys,

I just have a general question, How would you find the equation of a line with just the equations of the planes intersecting. I know for the direction vector it would be the cross product, but for the other part of the equation r=ro +tv, the ro part i do not know how to find.
 
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  • #2
[tex]r_0[/tex] is any single point on the line of intersection, which means you just need to find anyone point that lies on both planes.
 

1. What is the equation of a line of intersecting planes?

The equation of a line of intersecting planes can be found by determining the point of intersection between the two planes and using that point as the starting point for the equation. The direction vector of the line is then found by taking the cross product of the normal vectors of the two planes. The final equation will be in the form of r = a + td, where a is the point of intersection and d is the direction vector of the line.

2. How do I find the point of intersection between two planes?

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

3. What is the significance of the direction vector in the equation of a line of intersecting planes?

The direction vector in the equation of a line of intersecting planes represents the direction in which the line is moving. It is perpendicular to both of the normal vectors of the intersecting planes and helps to define the orientation of the line.

4. Can the equation of a line of intersecting planes be simplified?

Yes, the equation of a line of intersecting planes can be simplified by finding the unit vector of the direction vector and using that instead. This will result in a simpler equation of r = a + tu, where u is the unit vector in the direction of the line.

5. Is it possible for two planes to not intersect and have an equation of a line?

No, if two planes do not intersect, then there is no point of intersection and therefore no line can be formed. In order for a line to be formed, the two planes must intersect at a single point.

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