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The problem you provided consists of two tasks related to planes and their intersection in 3D space.

### Given planes:
1. \( y - z = 1 \)
2. \( x + y + 2z = 3 \)

These two planes intersect along a line \( \epsilon \).

### Task (a): Find the parametric and symmetric equations of the line \( \epsilon \).

To find the parametric and symmetric equations of the line formed by the intersection of two planes, we need to:
- Solve the system of equations.
- Find a point on the line.
- Find a direction vector for the line.

#### Step 1: Solve the system of equations

We have:
1. \( y - z = 1 \)
2. \( x + y + 2z = 3 \)

From the first equation:
\[
y = z + 1
\]

Substitute this into the second equation:
\[
x + (z + 1) + 2z = 3
\]
\[
x + 3z + 1 = 3
\]
\[
x = 2 - 3z
\]

So, the parametric form of the line is:
\[
x = 2 - 3t, \quad y = t + 1, \quad z = t
\]
where \( t \) is the parameter.

#### Parametric equations of the line:
\[
x = 2 - 3t, \quad y = t + 1, \quad z = t
\]

#### Step 2: Symmetric form

To express the symmetric form, eliminate the parameter \( t \) from the parametric equations:
\[
t = z
\]
From \( x = 2 - 3t \), we get:
\[
t = \frac{2 - x}{3}
\]
From \( y = t + 1 \), we get:
\[
t = y - 1
\]

Thus, the symmetric form is:
\[
\frac{2 - x}{3} = z = y - 1
\]

---

### Task (b): Find the equation of a third plane that contains the line \( \epsilon \) and is perpendicular to the plane \( x + y - 2z = 3 \).

To solve this, we follow these steps:

#### Step 1: Find the normal vector of the given plane.

The plane \( x + y - 2z = 3 \) has a normal vector

Τα επίπεδα y - z = 1 και x + 2z = 3 τέμνονται σε μία ευθεία ε. (α) Να βρείτε τις παραμετρικές και τις συμμετρικές εξισώσεις της ε. (β) Να βρείτε την εξίσωση ενός τρίτου επιπέδου που περιέχει την ε και είναι κάθετο στο επίπεδο x + y - 2z = 3.

Learn how to find the parametric and symmetric equations of a line formed by the intersection of two planes in 3D space. This problem covers key linear algebra concepts and includes finding the equation of a third plane perpendicular to a given one.

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