To solve the problem step by step:

Step 1: Find the equation of line $L$, the intersection of the two planes

The two planes are:

1. $x - z = 1$
2. $y + 2z = 3$

Parameterizing the line:

Express $x, y, z$ in terms of a parameter $t$.

• From $x - z = 1$: $x = z + 1$

• From $y + 2z = 3$: $y = 3 - 2z$

Let $z = t$ (parameterize $z$ as $t$): $x = t + 1, \quad y = 3 - 2t, \quad z = t$

Thus, the parametric equations of the line $L$ are: $x = t + 1, \quad y = 3 - 2t, \quad z = t$

Or in vector form: $\vec{r}(t) = (1, 3, 0) + t(1, -2, 1)$

Step 2: Plane $Q$ passes through $L$ and is perpendicular to $x + y - 2z = 1$

The plane $Q$ contains $L$, so its normal vector $\vec{n}_Q$ must be perpendicular to the line $L$. Also, $Q$ is perpendicular to the plane $x + y - 2z = 1$, so $\vec{n}_Q$ is parallel to the normal vector of this plane, $\vec{n}_1 = (1, 1, -2)$.

Normal vector to $Q$:

The direction vector of $L$ is $\vec{d}_L = (1, -2, 1)$. Since $Q$ contains $L$, $\vec{n}_Q$ must be orthogonal to $\vec{d}_L$: $\vec{n}_Q \cdot \vec{d}_L = 0$

Let $\vec{n}_Q = (a, b, c)$. Then: $a(1) + b(-2) + c(1) = 0$ $a - 2b + c = 0$

Additionally, $\vec{n}_Q$ is parallel to $\vec{n}_1 = (1, 1, -2)$, so: $\vec{n}_Q = k(1, 1, -2)$

Let $\vec{n}_Q = (1, 1, -2)$ (choosing $k = 1$).

Equation of plane $Q$:

The general equation of a plane is: $x + y - 2z = d$

To find $d$, substitute a point on $L$ (e.g., $(1, 3, 0)$): $1 + 3 - 2(0) = d$ $d = 4$

Thus, the equation of plane $Q$ is: $x + y - 2z = 4$

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Final Answers:

1. Equation of line $L$: $x = t + 1, \quad y = 3 - 2t, \quad z = t$ or: $\vec{r}(t) = (1, 3, 0) + t(1, -2, 1)$

2. Equation of plane $Q$: $x + y - 2z = 4$

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Would you like additional clarifications or examples?

Related Questions:

1. How do you find the intersection of two planes geometrically?
2. What is the condition for a plane to be perpendicular to another plane?
3. Can a line lie completely within two planes simultaneously? Why?
4. How do parameterized line equations relate to vector geometry?
5. What are some other forms of representing a plane equation?

Tip:

When solving problems involving planes and lines, always check for consistency by substituting values into the equations. This ensures the solution satisfies all given conditions.