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When dealing with planes in geometry, the term "normal vector" plays a crucial role. A normal vector is a vector that is perpendicular to a surface or a line. In the context of planes, this vector is perpendicular to any line lying on the plane.
For a plane, the normal vector provides a unique direction that is not parallel to any direction in the plane itself. It is vital because it helps determine the tilt or orientation of the plane.
To find the equation of a plane, the normal vector is often key. In our example, the normal vector \((0, 1, 0)\) indicates that our plane is parallel to the \(x\) and \(z\) dimensions, and perpendicular to the \(y\)-axis.

Parallel planes are two planes that do not intersect and remain the same distance apart across their entire length. They share the same orientation, meaning their normal vectors are either identical or scalar multiples of one another.
If you have a plane parallel to the \(xz\)-plane, it means its orientation is the same as the \(xz\)-plane. In our particular problem, the plane containing the point (1,2,3) is parallel to the \(xz\)-plane, which is equivalent to having a normal vector \((0, 1, 0)\).
Since the plane's normal vector is pointing along the \(y\)-axis, any movement in the plane does not change the \(y\)-coordinate. This is why the equation of our plane is simply \(y = 2\), denoting that everywhere on the plane, \(y\) remains constant at 2.

Coordinate geometry, often called analytic geometry, is a branch of geometry where points are defined by their positions on a plane using ordered pairs or triples in space.
This approach simplifies the study of geometric shapes by representing them with equations. It becomes easier to determine distances, angles, and other properties using algebraic methods.
In our example, the given point (1,2,3) in space helps us anchor the plane's position. The normal vector \((0, 1, 0)\) tells us the orientation, and coordinate geometry allows us to combine these to derive the plane’s equation, showing how powerful this technique can be in spatial analysis.