Determine the equation of the line o intersection between the planes

2x-2y+z-4=0 and 2x-3y-z+8=0
State as vector equation

add them:

4x - 5y + 4 = 0
4x = 5y - 4
let y = 0, then x = -1
plug into first equation: 2x-2y+z-4=0
-2 + 0 + z - 4 = 0 , z = 6
so you have the point (-1, 0, 6) on the line of intersection

let y = 4, then x = 4 and
8 - 8 + z - 4 = 0 , z = 4
and (4,4,4) is another point on the line of intersection.

You know have two points and you can find the vector equation of the line

To determine the equation of the line of intersection between the two planes, we need to find two points on that line. These points can be obtained by finding the solution to the system of equations formed by the two planes.

Let's start by forming the system of equations:

1) 2x - 2y + z - 4 = 0
2) 2x - 3y - z + 8 = 0

To eliminate one variable, we can add the two equations together:

3) (1) + (2): (2x - 2y + z - 4) + (2x - 3y - z + 8) = 0

Simplifying equation (3), we get:

4) 4x - 5y + 4 = 0

Now, let's solve the system of equations formed by (1) and (4). Rearranging equation (4) to isolate x:

5) 4x = 5y - 4
6) x = (5y - 4) / 4

Substituting equation (6) into equation (1), we can solve for z:

7) 2((5y - 4) / 4) - 2y + z - 4 = 0

Simplifying equation (7), we get:

8) z = (11y - 12) / 4

Now, we have expressions for x and z in terms of y. So, the vector equation for the line of intersection between the two planes is:

r = <(5y - 4) / 4, y, (11y - 12) / 4>, where y is a parameter.

Thus, the equation of the line of intersection between the two planes is: r = <(5y - 4) / 4, y, (11y - 12) / 4>, with y as a parameter.