Finding the solution of three planes

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In summary, the planes with equations x + y + 7z = -7, 2x + 3y + 17z = -16, and x + 2y + (a^2 + 1) z = 3a cannot all have the same solution, and there will be multiple solutions (points along a line) if the last equation is 0z = k, with k not equal to 0. Finally, there will be a unique solution (a single point) if the last equation is kz = <whatever>, with k not equal to 0.
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Homework Statement


given the planes with equations:
x + y + 7z = -7
2x + 3y + 17z = -16
x + 2y + (a^2 + 1) z = 3a

find values for the constant a for which:
-there are no solutions
-the planes meet in a line. in this case find the parametric equation of the line
-meet at a point. then find the parametric equation of the point.

Homework Equations



gaussian elimination? I don't think there are any. =S

The Attempt at a Solution



I put the planes into an augmented matrix, and then into echelon form, and got (I think its correct) the general solution, which is attached. From the I can read off the equation of the line, right? The parts I'm really stuck on are parts one and three. I realize that for the planes to meet in a point, there has to be 3 pivots but beyond that I'm a bit stumped.

Well actually, I have a bit of a clue for part one, that a =/= 3 because then 0z = 18 which is inconsistent. is that right? But for part 3, I'm not sure how to do it, really.
 

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  • #2
gracelette said:

Homework Statement


given the planes with equations:
x + y + 7z = -7
2x + 3y + 17z = -16
x + 2y + (a^2 + 1) z = 3a

find values for the constant a for which:
-there are no solutions
-the planes meet in a line. in this case find the parametric equation of the line
-meet at a point. then find the parametric equation of the point.

Homework Equations



gaussian elimination? I don't think there are any. =S

The Attempt at a Solution



I put the planes into an augmented matrix, and then into echelon form, and got (I think its correct) the general solution, which is attached. From the I can read off the equation of the line, right? The parts I'm really stuck on are parts one and three. I realize that for the planes to meet in a point, there has to be 3 pivots but beyond that I'm a bit stumped.

Well actually, I have a bit of a clue for part one, that a =/= 3 because then 0z = 18 which is inconsistent. is that right? But for part 3, I'm not sure how to do it, really.
You have a mistake. In your first augmented matrix, the bottom row is 0 0 a2 - 9 | 3a + 9

Right after that, you say let a = -3i. There are two real values that make a2 - 9 equal to 0.

For your other questions, there will be no solutions if the last equation is 0z = k, with k not equal to 0.
There will be multiple solutions (points along a line) if the last equation is 0z = 0.
Finally, there will be a unique solution (a single point) if the last equation is kz = <whatever>, with k not equal to 0.
 
  • #3
Ooops. I guess this is what happens when I try do maths at obscene times of day. Thankyou, i worked it out now!
 

FAQ: Finding the solution of three planes

1. What is the solution of three planes?

The solution of three planes refers to the point at which all three planes intersect or meet. In other words, it is the point that satisfies the equations of all three planes simultaneously.

2. How do you find the solution of three planes?

To find the solution of three planes, you can use a method called Gaussian elimination. This involves reducing the three equations into a simpler form, known as row-echelon form, and then solving for the variables using back substitution.

3. Can there be more than one solution for three planes?

Yes, there can be multiple solutions for three planes. If the three planes are parallel or coincide with each other, there will be infinitely many solutions. However, if the three planes are not parallel, there will be either one unique solution or no solution at all.

4. What does it mean if there is no solution for three planes?

If there is no solution for three planes, it means that the three planes do not intersect at any point. In other words, the equations of the three planes are inconsistent and cannot be satisfied simultaneously.

5. How is finding the solution of three planes useful in real life?

Finding the solution of three planes is useful in many fields such as engineering, physics, and computer graphics. It can be used to determine the intersection point of three objects or surfaces, which is important in designing and constructing structures or creating 3D models. For example, in engineering, the solution of three planes can be used to determine the coordinates of a point where beams or pipes intersect.

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