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When working with planes and their equations, the normal vector plays a crucial role. A normal vector is a vector that is perpendicular to a surface. In context, this means the vector is perpendicular to the plane.

• For a given plane with a general equation: \[ Ax + By + Cz = D \] where the coefficients \(A, B,\) and \(C\) are the components of the normal vector.
• The normal vector provides directionality and is essential in defining the orientation of the plane.

In the given exercise, since the plane of interest is parallel to the \(y\)-plane, it shares the same normal vector as the \(y\)-plane. Therefore, the normal vector is \((0,1,0)\). This vector indicates the plane is vertical and runs parallel to the \(y\)-axis with no inclination along the \(x\) and \(z\) axes.

Parallel planes are two or more planes that never intersect. They have the same orientation in space. Practically, this means they will either have identical normal vectors or their normal vectors will be scalar multiples of one another.

• When two planes are parallel, the distance between them is uniform and constant.
• The mathematical representation asserts that parallel planes maintain the same normal vector, ensuring they remain equidistant and never meet.

In the given problem, the plane through \((2,4,2)\) is parallel to the \(y\)-plane. Thus, it shares its normal vector, \((0,1,0)\), ensuring they do not intersect. This shared normal vector is central in determining the plane equation so it remains parallel.

Coordinate geometry, also known as analytic geometry, allows us to use algebraic techniques to solve geometric problems. It involves understanding and interpreting geometric figures like points, lines, and planes within a coordinate system.

• By utilizing coordinates, we can express the position of points and directions of lines and planes numerically.
• This makes it possible to formulate equations that describe these geometric forms.

The problem requires identifying a plane through a point \((2,4,2)\) using the coordinate geometry framework. The task involves working with normal vectors and finding equations based on given coordinates.
This approach ensures precise formulation and clear definitions of geometric entities such as the derived equation \(y = 4\). Analytic geometry thus bridges algebra and geometry, enabling us to visualize and solve planar problems efficiently.