Let's analyze the given statements one by one:

1. "Скалярное произведение нормальных векторов плоскостей, принадлежащих одной связке плоскостей, всегда отлично от нуля."

   • This statement is incorrect. If two planes belong to the same "family" of planes, their normal vectors might be parallel. In such a case, their scalar product could be zero.

2. "Плоскость $2x + 4y - z + 8 = 0$ пересекает ось $Ox$ в точке $(-4; 0; 0)$."

   • To find the intersection of this plane with the $Ox$ axis, set $y = 0$ and $z = 0$ and solve for $x$: $2x + 8 = 0 \quad \Rightarrow \quad x = -4.$ Thus, the point of intersection is $(-4, 0, 0)$, making this statement correct.

3. "Плоскость $2y - 1 = 0$ параллельна координатной плоскости $XOZ$."

   • The equation of the plane simplifies to $y = \frac{1}{2}$, which is a plane parallel to the $XOZ$ plane (since it only shifts along the $y$-axis). Hence, this statement is correct.

4. "Уравнения $2x + y - z + 1 = 0$ и $2x + y + z - 1 = 0$ определяют одну и ту же плоскость."

   • To determine if these two equations represent the same plane, subtract the second equation from the first: $(2x + y - z + 1) - (2x + y + z - 1) = 0,$ which simplifies to: $-2z + 2 = 0 \quad \Rightarrow \quad z = 1.$ Since this leads to $z = 1$, it shows that the two equations represent different planes. Therefore, this statement is incorrect.

Summary of Correct Statements:

• The second statement about the plane intersecting the $Ox$ axis at $(-4, 0, 0)$.
• The third statement about the plane $2y - 1 = 0$ being parallel to the $XOZ$ plane.

Would you like more details or have any further questions?

Here are 5 related questions:

1. How can you determine if two planes are parallel?
2. What does the scalar product of two normal vectors represent geometrically?
3. How do you find the intersection of a plane with a specific axis?
4. What is the relationship between the equations of two parallel planes?
5. How can we verify if two planes are identical based on their equations?

Tip: When checking if two planes are parallel, their normal vectors should be scalar multiples of each other.