Find the intersection of three planes (a line)?

In summary, the given system of equations represents three planes that intersect in a line. The task is to determine the values of p and q. The solution involves solving for the variables x, y, and z and choosing specific values for the parameters s and q to find a solution that is not unique.
  • #1
lillybeans
68
1

Homework Statement



The following system of equations represents three planes that intersect in a line.

1. 2x+y+z=4
2. x-y+z=p
3. 4x+qy+z=2

Determine p and q

2. The attempt at a solution

The problem I have with this question is that you are solving 5 variables with only 3 equations. I attempted at this question for a long time, to no avail.

What I did was I tried to convert everything into parametric form, so to make z a parameter (s) and express x and y in terms of s. The steps are as follows:

1. Let z=s

In terms of p and s
2. From ① + ②, we get ④ x=(4+p-2s)/3 ---> "y" eliminated
3. From ① - ②x2, we get ⑤ y=(4-2p+s)/3 ---> "x" eliminated

In terms of q and s
4. From ①x2 - ③, we get ⑥ y=(6-2)/(2-q) ---> "x" eliminated
5. From ①xq - ③, we get ⑦ x=(3q-2+s)/2(q-2) ---> "y" eliminated

I did not do operations with ② and ③, because that will re-introduce the 5 variables (x,y,z,p,q), and after elimination there will still be 4, which is not what we want. We want as little as variables as possible and eliminate as many as possible.

Since they all intersect at the same line, I can make ④=⑦, ⑤=⑥. That's 2 equations. Since we are now down to 3 variables (p,q,s), we need a 3rd equation containing these 3 variables to solve. I was thinking of bringing back ② and ③, but that doesn't work because I will be re-introducing either x or y after elimination.

Please help me and correct my thought process if anywhere along the way I didn't seem to make much sense.

Thank you all so much!
 
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  • #2
Treat it as three equations in three unknowns, x, y, and z, letting s and q be the parameters. I can't see any reason to think that z= s.

Just go ahead and solve the equations for x, y, and z, leaving s and q as numbers in the solution. Then choose s and q so that there exist a solution (the planes are not parallel) but there is no unique solution (the planes do not all cross at a single point).

For example, it is easy see that we an eliminate y by adding the first two equations:
3x+ 2z= p+ 4. We can also eliminate y by multiplying the second equation by q and adding to the third equation: (q+ 4)x+ (q+ 1)z= pq+ 2.

Now, say, multiply the first equation by q+ 1, the second equation by 2, and subtract to eliminate z. The gives a solution for x depending upon q and s. For what values of q and s can you not get a single solution?

Remember that 0x= a has no solution but that 0x= 0 has many solutions.
 
  • #3
HallsofIvy said:
Treat it as three equations in three unknowns, x, y, and z, letting s and q be the parameters. I can't see any reason to think that z= s.

Just go ahead and solve the equations for x, y, and z, leaving s and q as numbers in the solution. Then choose s and q so that there exist a solution (the planes are not parallel) but there is no unique solution (the planes do not all cross at a single point).

For example, it is easy see that we an eliminate y by adding the first two equations:
3x+ 2z= p+ 4. We can also eliminate y by multiplying the second equation by q and adding to the third equation: (q+ 4)x+ (q+ 1)z= pq+ 2.

Now, say, multiply the first equation by q+ 1, the second equation by 2, and subtract to eliminate z. The gives a solution for x depending upon q and s. For what values of q and s can you not get a single solution?

Remember that 0x= a has no solution but that 0x= 0 has many solutions.

Thank you so much for your help! I got it. (q=5)
 

What does it mean to find the intersection of three planes?

When we talk about finding the intersection of three planes, we are looking for the point or line where all three planes intersect. In other words, it is the solution to a system of three linear equations in three variables.

Why is finding the intersection of three planes important?

Finding the intersection of three planes is important in many fields, such as engineering, physics, and computer graphics. It allows us to determine the point or line where three objects in space intersect, which can be useful for solving real-world problems or creating visual representations.

What are the steps involved in finding the intersection of three planes?

The first step is to write out the equations for each of the three planes. Then, we can use algebraic methods such as substitution or elimination to solve the system of equations and find the values for the variables. Finally, we can use these values to determine the coordinates of the intersection point or the equation of the intersection line.

Can there be more than one intersection point or line for three planes?

Yes, it is possible for three planes to have more than one intersection point or line. This occurs when the planes are parallel or coincide with each other. In these cases, the system of equations will have either infinitely many solutions or no solution at all.

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

Finding the intersection of three planes has many practical applications, such as determining the point of impact in a three-dimensional collision, calculating the trajectory of a projectile, and creating 3D models in computer graphics. It is also used in fields like architecture, where it can help determine the best placement of beams and supports in a building.

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