To find the parametric equations for the line of intersection of the planes x+y+z=3 and x-y+2z=2, there are a few steps missing in your procedure. Let's go through the correct steps:
1. Start with the two given equations for the planes:
- Plane 1: x + y + z = 3
- Plane 2: x - y + 2z = 2
2. To find the line of intersection, we need to solve the system of equations formed by the two planes. We can use the method of substitution or elimination to do this.
3. Let's use the elimination method to eliminate x. To do so, subtract the second equation from the first equation:
- (x + y + z) - (x - y + 2z) = 3 - 2
- x - x + y - (-y) + z - 2z = 1
- 2y - z = 1
4. Now, we have two equations:
- y + z = 3 (Equation 1)
- 2y - z = 1 (Equation 2)
5. Solve the system of equations (Equation 1 and Equation 2) to find the values of y and z. To do so, you can either use substitution or elimination. For simplicity, let's use the elimination method:
- Multiply Equation 2 by 2:
4y - 2z = 2
- Add Equation 1 and the modified Equation 2:
(y + z) + (4y - 2z) = 3 + 2
5y - z = 5
- Now, you have a new equation:
5y - z = 5 (Equation 3)
6. We now have two equations:
- 2y - z = 1 (Equation 2)
- 5y - z = 5 (Equation 3)
7. Solve the new system of equations using the elimination method:
- Multiply Equation 2 by 5 and Equation 3 by 2:
10y - 5z = 5
10y - 2z = 10
- Subtract the modified Equation 2 from the modified Equation 3:
(10y - 2z) - (10y - 5z) = 10 - 5
3z = 5
- Solve for z:
z = 5/3
8. Substitute the value of z back into either Equation 2 or Equation 3 to solve for y. Let's use Equation 2:
2y - (5/3) = 1
2y = 1 + 5/3
2y = 8/3
y = (8/3)(1/2)
y = 4/3
9. We now have the values of y and z. Substitute these values into one of the original equations (let's use Equation 1) to solve for x:
y + z = 3
(4/3) + (5/3) = 3
x = 3 - (4/3) - (5/3)
x = (9/3) - (4/3) - (5/3)
x = 0
10. The point of intersection of the two planes is (0, 4/3, 5/3).
11. To find the parametric equations for the line of intersection, let t be a parameter:
x = 0 + 0t = 0
y = (4/3) + (1/3)t
z = (5/3) + (1/3)t
Therefore, the correct parametric equations for the line of intersection are:
x = 0
y = (4/3) + (1/3)t
z = (5/3) + (1/3)t
Note: The equation you obtained, 3i - j - 2k, is the direction vector of the line of intersection, but it is not the parametric equation itself.