Understanding how a plane is uniquely determined in three-dimensional (3D) space is a fundamental concept in geometry. A plane is a flat, two-dimensional surface that extends infinitely. For a plane to be uniquely determined, there must be specific conditions that define one and only one such plane.

Plane Determination Basics

Several standard geometric conditions are known to uniquely determine a plane. Let's analyze the commonly accepted methods for unique plane determination:

• A Normal to the Plane and its Distance from the Origin: If we know the direction perpendicular to the plane (its normal vector) and the shortest distance from the origin (0,0,0) to the plane, the plane is uniquely defined. This is often represented by the normal form of the plane equation: \(\vec{r} \cdot \hat{n} = d\), where \(\vec{r}\) is the position vector of any point on the plane, \(\hat{n}\) is the unit normal vector, and \(d\) is the perpendicular distance from the origin to the plane.
• Three Non-Collinear Points: Any three points that do not lie on the same straight line (non-collinear) will always define one and only one plane. Imagine holding three fingers (not in a line) and trying to place a flat surface on them; there's only one way to do it. This method provides enough information to fix a unique plane.
• A Point and Perpendicular Direction: If a plane passes through a specific point and its normal (perpendicular) direction is given, the plane is uniquely determined. This is because the given direction serves as the normal vector to the plane, fixing its orientation, and the given point anchors it in space. The equation can be written as \(A(x - x_1) + B(y - y_1) + C(z - z_1) = 0\), where \((x_1, y_1, z_1)\) is the given point and \((A, B, C)\) is the normal vector defining the perpendicular direction.

Option Analysis for Plane Determination

Let's examine each given option in the context of uniquely determining a plane:

• Option 1: The normal to the plane and its distance from the origin is given.

  As discussed, this is a standard and well-defined method to uniquely determine a plane. Knowing the normal vector fixes the plane's orientation, and the distance from the origin fixes its position in 3D space. This condition indeed defines a unique plane.

• Option 2: It passes through three given non collinear points.

  This is another fundamental and commonly used way to uniquely define a plane. Three points that are not collinear provide enough spatial information to fix a single, unique flat surface.

• Option 3: It passes through a point and is perpendicular to a given direction.

  This condition also uniquely determines a plane. The given direction serves as the normal vector to the plane, establishing its orientation, and the point ensures its specific location in space. This is a common way to define a plane equation.

• Option 4: The equation of a plane lying in extraordinary form.

  The term "extraordinary form" is not a recognized or standard mathematical form for the equation of a plane in geometry. Standard forms include the general form (\(Ax + By + Cz + D = 0\)), intercept form (\(\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\)), normal form, and point-normal form. Since "extraordinary form" is not a defined mathematical concept in this context, it cannot be a "known factor" that uniquely determines a plane. If a form is not defined or understood, it cannot provide the necessary information for unique determination of a plane.

Unique Plane Determination Conclusion

Based on the analysis, the options that are known factors to uniquely determine a plane are clearly defined in geometry. The option that does not fit this criterion is the one that refers to an undefined concept.

Therefore, the statement "The equation of a plane lying in extraordinary form" is NOT a known factor to uniquely determine a plane because it refers to a non-standard or undefined mathematical concept in this context of 3D geometry.