Final answer:
A plane requires at least three noncollinear points to be defined. In two dimensions, vectors have two components and in three dimensions, they have three components, corresponding to the x, y, and z axes. The parallelogram rule is used to add two vectors in a plane geometrically.
Step-by-step explanation:
The question revolves around the basic principles of geometry and vector analysis. Pointedly, it explores how points, lines, planes, and vectors interact within two-dimensional and three-dimensional spaces.
A plane is a flat, two-dimensional surface that extends infinitely in all directions. It's a foundational concept in geometry that contains at least three noncollinear points. Noncollinear points are points that do not all lie on the same straight line. Once you have three noncollinear points, you can determine a unique plane in which these points lie. The statement 'Three noncollinear points would have at least two lines containing them' is incorrect because only one line can pass through any two points, which means it's impossible for two lines to contain all three noncollinear points at once.
The creation of vectors in different dimensions demands different numbers of components: two for a plane and three for space. When discussing vectors on a plane, we can visualize this through the parallelogram or tail-to-head rule for the addition of vectors. For example, to construct a resultant vector from two vectors in a plane, we can apply the parallelogram rule to determine the vector's magnitude and direction.
In a three-dimensional context, a point's location is specified by three coordinates (x, y, z), where x and y define a position within a plane and z indicates the vertical displacement from that plane. This is an extension of the two-dimensional Cartesian coordinate system, which uses just the x and y axes.