Finding the Line of Intersection for Two Planes

In summary, to find the line of intersection of the two planes 7x - 2y + 3z = -2 and -3x + y + 2z + 5 =0 without using row reduction, you can calculate the cross product of the two normal vectors and plug in one point of intersection. Alternatively, you can solve the two equations for two variables and leave the third as a parameter. The resulting line of intersection can be expressed as x= -7t-12, y= -23t-41, z=t.
  • #1
theorist
1
0
Hello,

How do I find the line of intersections of the two planes 7x - 2y + 3z = -2 and -3x + y + 2z + 5 =0, without having to resort to solving it by row reduction?
 
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  • #2
Calculate the cross product of the two normal vectors of the planes. Then plug in one point of intersection to yield the line of intersection.

cookiemonster
 
  • #3
Or, if that is too much trouble, solve the two equations for two of the variables, leaving the third as parameter: from 7x - 2y + 3z = -2 and -3x + y + 2z + 5 =0, multiply the second equation by 2 and add to get
(7+2(-3))x+ (-2+2)y+ (3+2(2))z+ 2(5)= -2 or x+ 7z+ 10= -2. From that,
x= -7z- 12 and then, using the second equation, -3(-7z-12)+ y+ 2z+ 5= 0 so
y= -23z- 41 or, writing the parameter as "t"
x= -7t- 12, y= -23t- 41, z= t is the line of intersection.
 

What is plane-plane intersection?

Plane-plane intersection refers to the mathematical concept of finding the points of intersection between two planes in three-dimensional space. It is commonly used in various fields of science and engineering, such as computer graphics, robotics, and physics.

How is plane-plane intersection calculated?

The calculation of plane-plane intersection involves finding the equations of the two planes and then solving them simultaneously to determine the coordinates of the points of intersection. This can be done using various methods, such as substitution or elimination.

What are the applications of plane-plane intersection?

Plane-plane intersection has many practical applications, including determining the path of a moving object in three-dimensional space, finding the intersection of two roads or runways, and calculating the angle between two planes.

What are some challenges or limitations of plane-plane intersection?

One major challenge of plane-plane intersection is that it can be computationally expensive, especially when dealing with complex shapes or large datasets. Another limitation is that it assumes the planes are perfect and do not have any curvature or irregularities.

How is plane-plane intersection related to other mathematical concepts?

Plane-plane intersection is closely related to other mathematical concepts, such as linear algebra, geometry, and graph theory. It also has applications in fields like computer vision, where it is used to analyze and interpret images in three-dimensional space.

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