To find the equation of a plane that passes through the line of intersection of two given planes and is also perpendicular to a third plane, we follow these steps:

1. General form of a plane through the intersection of two planes

Given the planes:

$P_1: x - z = 2$ $P_2: y + 4z = 2$

The equation of the plane passing through their line of intersection can be written as a linear combination of these two planes:

$\text{Plane equation: } (x - z - 2) + \lambda(y + 4z - 2) = 0$

Here, $\lambda$ is a scalar parameter.

2. Perpendicular condition to the plane $x + y - 4z = 4$

The normal vector of the plane we want to be perpendicular to is the normal vector of the plane $x + y - 4z = 4$, which is $\mathbf{n}_3 = (1, 1, -4)$.

The normal vector of the plane we're trying to find will be the combination of the normal vectors of the two planes:

• Normal vector of $P_1: x - z = 2$ is $\mathbf{n}_1 = (1, 0, -1)$
• Normal vector of $P_2: y + 4z = 2$ is $\mathbf{n}_2 = (0, 1, 4)$

Thus, the normal vector of the desired plane is:

$\mathbf{n} = \mathbf{n}_1 + \lambda \mathbf{n}_2 = (1, 0, -1) + \lambda(0, 1, 4) = (1, \lambda, -1 + 4\lambda)$

For the plane to be perpendicular to the plane $x + y - 4z = 4$, the dot product of the normal vectors $\mathbf{n}$ and $\mathbf{n}_3 = (1, 1, -4)$ must be zero:

$\mathbf{n} \cdot \mathbf{n}_3 = (1, \lambda, -1 + 4\lambda) \cdot (1, 1, -4) = 1 \cdot 1 + \lambda \cdot 1 + (-1 + 4\lambda) \cdot (-4) = 0$

3. Solve for $\lambda$

Expanding the dot product:

$1 + \lambda + (-1 + 4\lambda)(-4) = 0$ $1 + \lambda + 4 - 16\lambda = 0$ $5 - 15\lambda = 0$ $\lambda = \frac{5}{15} = \frac{1}{3}$

4. Equation of the plane

Now substitute $\lambda = \frac{1}{3}$ into the equation of the plane:

$(x - z - 2) + \frac{1}{3}(y + 4z - 2) = 0$

Multiplying everything by 3 to eliminate the fraction:

$3(x - z - 2) + (y + 4z - 2) = 0$ $3x - 3z - 6 + y + 4z - 2 = 0$ $3x + y + z - 8 = 0$

Thus, the equation of the required plane is:

$\boxed{3x + y + z = 8}$

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Would you like further clarification on any step or additional details?

Here are 5 related questions to explore:

1. How do you find the intersection line of two planes?
2. What is the geometric meaning of the dot product between two vectors?
3. How can you check if a line lies on a plane?
4. Can two planes be perpendicular to each other? How would you prove it?
5. What are the conditions for a plane to be parallel to another plane?

Tip: The normal vector of a plane is a key element in understanding the plane's orientation in space, and it helps to solve problems related to perpendicularity and parallelism.