Give a geometrical interpretation of the intersection of the planes with equations
x + y + z + 3 = 0
2x + 3y - 6z - 1 = 0
4x + 6y -12z + 11 = 0
The intersection of three planes in three-dimensional space is a point, a line, or empty (no intersection).
To find the intersection of these three planes, we can solve the system of equations formed by the three equations.
To do so, we can use the method of Gaussian elimination or any other method of solving systems of linear equations.
After solving the system of equations, we find that the intersection of the planes is a single point, which can be expressed as (x, y, z) = (-4, 5, -2).
Geometrically, this means that the three planes intersect at a single point (-4, 5, -2) in three-dimensional space.
To find the geometrical interpretation of the intersection of the planes, we need to first determine the nature of the solution for the system of equations formed by the planes.
1. Start by writing the system of equations in matrix form:
| 1 1 1 | | x | | -3 |
| 2 3 -6 | * | y | = | 1 |
| 4 6 -12 | | z | | -11 |
2. Convert the augmented matrix into its row-echelon form (REF) or reduced row-echelon form (RREF) using elementary row operations such as row swapping, multiplication, or addition. This will help us determine the nature of the solution.
| 1 1 1 | | x | | -3 |
| 0 1 -8 | * | y | = | -7 |
| 0 0 0 | | z | | 0 |
3. Analyzing the RREF, we can see that the third row corresponds to the equation 0z = 0, which means that z can have any value. This implies that the planes are parallel in the z-direction.
4. Looking at the second row, we have 0y = -7, which is a contradiction since no real number multiplied by 0 can give a non-zero value. This means that the system of equations has no solution in the y-direction, and the planes are also parallel in the y-direction.
5. Finally, since we have two parallel planes, the intersection occurs in a line.
Therefore, the geometrical interpretation of the intersection of the given planes is a line.
To find the geometrical interpretation of the intersection of the planes with the given equations, we can start by solving the system of equations.
Equation 1: x + y + z + 3 = 0
Equation 2: 2x + 3y - 6z - 1 = 0
Equation 3: 4x + 6y - 12z + 11 = 0
To solve this system, we can use the method of Gaussian elimination or matrix operations. Here, we will use Gaussian elimination:
Step 1: We can subtract 2 times Equation 1 from Equation 2 to eliminate the x coefficient:
(2x + 3y - 6z - 1) - 2(x + y + z + 3) = 0
2x + 3y - 6z - 1 - 2x - 2y - 2z - 6 = 0
y - 8z - 7 = 0
Step 2: We can subtract 4 times Equation 1 from Equation 3 to eliminate the x coefficient:
(4x + 6y - 12z + 11) - 4(x + y + z + 3) = 0
4x + 6y - 12z + 11 - 4x - 4y - 4z - 12 = 0
2y - 16z - 1 = 0
Now, we have the following two equations:
Equation 4: y - 8z - 7 = 0
Equation 5: 2y - 16z - 1 = 0
By combining Equations 4 and 5, we can find the value of y in terms of z:
y = 8z + 7
Substitute this value back into Equation 5:
2(8z + 7) - 16z - 1 = 0
16z + 14 - 16z - 1 = 0
13 = 0
However, 13 does not equal 0, which means there is no valid solution for this system of equations. Therefore, the planes represented by these equations do not intersect at any common point or form a common line.
The geometrical interpretation of this is that the three planes are parallel to each other and do not intersect.