To determine how a point is represented in a three-dimensional Cartesian coordinate system, we need to understand the standard notations used in such a system.
A Cartesian coordinate system in three dimensions uses three axes, commonly denoted as the [tex]\(x\)[/tex]-axis, [tex]\(y\)[/tex]-axis, and [tex]\(z\)[/tex]-axis. These axes intersect at the origin [tex]\((0, 0, 0)\)[/tex], and each point in space is defined by an ordered triplet [tex]\((x, y, z)\)[/tex].
Let's examine each of the given options:
A. [tex]\((x, y, z)\)[/tex]
- This represents a point in space using three coordinates: [tex]\(x\)[/tex] for the position along the [tex]\(x\)[/tex]-axis, [tex]\(y\)[/tex] for the position along the [tex]\(y\)[/tex]-axis, and [tex]\(z\)[/tex] for the position along the [tex]\(z\)[/tex]-axis. This is the correct standard notation for a point in a three-dimensional Cartesian coordinate system.
B. [tex]\((w, x, y)\)[/tex]
- This notation does not follow the standard convention for a three-dimensional Cartesian coordinate system. Here, the variables [tex]\(w, x,\)[/tex] and [tex]\(y\)[/tex] are used, which is unconventional.
C. [tex]\(x, y, z\)[/tex]
- This option does not use parentheses to group the coordinates into a single point, which is necessary in the notation of a point in Cartesian coordinates.
D. [tex]\((x, y, y)\)[/tex]
- This notation repeats the [tex]\(y\)[/tex] coordinate and neglects [tex]\(z,\)[/tex] which is incorrect for a Cartesian coordinate system as it requires unique identifiers for the [tex]\(x, y,\)[/tex] and [tex]\(z\)[/tex] axes.
Given the options, the correct representation of a point in a three-dimensional Cartesian coordinate system is:
A. [tex]\((x, y, z)\)[/tex]
