Finding intersection of three planes

In summary: Thanks for the correction! In summary, the three given vectors represent three planes that intersect at three parallel lines. However, if the third vector is changed to (1, -2, -4), the planes will intersect at a single point. This is because the original third vector can be written as a linear combination of the other two, while the new vector cannot. This difference is due to the determinant of the coefficient matrix being 0 in the original case, resulting in a non-unique solution, while it is non-zero in the new case, resulting in a unique solution.
  • #1
Tarrius
1
0
Hi!

I'm having trouble with this question, any help would be much appreciated! :)

Q1: Given the three vectors:

n1 = (1, 2, 3)
n2 = (3, 2, 1)
n3 = (1, −2, −5)

Find the intersection of the three planes ni*x = 0. What happens if n3 = (1, −2, −4)? Why is this different?
 
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  • #2
Are you familiar with vector spaces and matrix rank?
 
  • #3
Tarrius said:
Hi!

I'm having trouble with this question, any help would be much appreciated! :)

Q1: Given the three vectors:

n1 = (1, 2, 3)
n2 = (3, 2, 1)
n3 = (1, −2, −5)

Find the intersection of the three planes ni*x = 0. What happens if n3 = (1, −2, −4)? Why is this different?
Your original n3=(1,-2,-5) = n2-2n1 so the three planes don't intersect. The pairwise intersections (of the planes) are 3 parallel lines.

The other n3 would work since it is not a linear combination of the others.
 
  • #4
mathman said:
Your original n3=(1,-2,-5) = n2-2n1 so the three planes don't intersect.
Actually they do intersect, just not in a single point as can be shown by the Rouché-Capelli theorem.
 
  • #5
You don't really need to know linear algebra- just the basics of systems of equations.
The planes defined by the first three vectors are
x+ 2y+ 3z= 0
3x+ 2y+ z= 0
x- 2y- 5z= 0.

Find the general solution to that system (there is NOT a unique solution because the determinant of coefficients is 0). What does that define, geometrically. The second set of equations do NOT have 0 determinant so have a unique solution. What that solution is should be obvious.
 
  • #6
da_nang said:
Actually they do intersect, just not in a single point as can be shown by the Rouché-Capelli theorem.
I forgot that the three parallel lines could be coincident.
 

FAQ: Finding intersection of three planes

What is the definition of finding intersection of three planes?

The intersection of three planes is the point or line where all three planes intersect or meet. It is the common solution to the equations of the three planes.

What are the methods used to find the intersection of three planes?

The two most commonly used methods are the substitution method and the elimination method. In the substitution method, you isolate one variable in one of the equations and substitute it into the other two equations to solve for the remaining variables. In the elimination method, you use algebraic operations to eliminate one variable at a time until you are left with one equation and one unknown variable, which can then be solved.

How many solutions can be obtained when finding the intersection of three planes?

There are three possible outcomes when finding the intersection of three planes: one unique solution, no solution, or infinitely many solutions. A unique solution occurs when the three planes intersect at a single point, a no solution occurs when the planes are parallel and do not intersect, and infinitely many solutions occur when the three planes are coincident and all intersect each other.

What are some real-life applications of finding the intersection of three planes?

Finding the intersection of three planes is used in many fields, such as engineering, physics, and computer graphics. It can be used to determine the location of a point in 3D space, the intersection of three roads or flight paths, and the position of a satellite in orbit. It is also used in computer graphics to create 3D images and objects.

What are some challenges in finding the intersection of three planes?

One of the main challenges is ensuring that the equations of the three planes are consistent and not contradictory. This can be done by checking for errors in the equations or by graphing the planes to visually see if they intersect. Another challenge is the potential for complex numbers to arise in the solutions, which may require further simplification or interpretation in real-life applications.

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