Final answer:
A point in a Cartesian system is represented by coordinates, a line can be seen as a point moving through space, and a plane is a flat, infinite two-dimensional surface. Vectors have x- and y-components in two dimensions and include a z-component in three dimensions, often illustrated using orthogonal unit vectors. The Cartesian coordinate system is used in both two-dimensional and three-dimensional spaces.
Step-by-step explanation:
Understanding Points, Vectors, and Planes in Cartesian Systems
In mathematics, the representation of a point, a line, and a plane can be described within the context of a Cartesian coordinate system. A point is represented by a pair of coordinates (x, y) in a plane and (x, y, z) in three-dimensional space. The point indicates a specific location within the system.
A line can be visualized as a point moving through space, creating a continuous path with no width. Mathematically, a line may be described by an equation in either two or three dimensions. In art, a line establishes a figure-ground relationship upon the surface it is drawn and can be used to create various forms.
When it comes to a plane, it is a flat, two-dimensional surface extending infinitely in all directions. A plane in a three-dimensional coordinate system may be defined by a normal vector and a point within the plane.
Vectors are important tools in representing lines and directions within these systems. In two dimensions, vectors are described by x- and y-components, and in three dimensions, by x-, y-, and z-components. The components are orthogonal projections onto the respective axes, following the parallelogram rule for vector addition. Moreover, in three dimensions, we use three orthogonal unit vectors, typically labeled i, ĵ and Ê, to define orientation within the system.
The Cartesian coordinate system is often employed in graphs and mappings because of its straightforward visualization on flat surfaces, such as paper. However, for three-dimensional analysis, a z-coordinate is added, which describes the elevation above or below the plane of the first two coordinates (x and y).
In geometry, a Cartesian coordinate system (UK: /kɑːrˈtiːzjən/, US: /kɑːrˈtiʒən/) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, called coordinate lines, coordinate axes or just axes (plural of axis) of the system. The point where they meet is called the origin and has (0, 0) as coordinates.