To answer the question of how to represent a point in a three-dimensional Cartesian coordinate system, we need to understand the standard notation used in mathematics for describing points in different dimensions.
In a two-dimensional Cartesian coordinate system, a point is represented by a pair of coordinates [tex]\((x, y)\)[/tex]. Here:
- [tex]\(x\)[/tex] is the horizontal coordinate (often referred to as the abscissa).
- [tex]\(y\)[/tex] is the vertical coordinate (often referred to as the ordinate).
When we extend this concept to three dimensions, we include an additional coordinate to represent the third dimension, usually denoted by [tex]\(z\)[/tex]. Thus, a point in a three-dimensional Cartesian coordinate system is represented by an ordered triple [tex]\((x, y, z)\)[/tex], where:
- [tex]\(x\)[/tex] is the coordinate along the x-axis (horizontal).
- [tex]\(y\)[/tex] is the coordinate along the y-axis (vertical).
- [tex]\(z\)[/tex] is the coordinate along the z-axis (depth, or the third dimension).
Now, let's evaluate the given options:
A. [tex]\((x, y, z)\)[/tex] - This correctly represents a point in three-dimensional space with the standard notation [tex]\(x, y, z\)[/tex].
B. [tex]\(x, y, z\)[/tex] - Although it uses the correct variables, the lack of parentheses makes it unsuitable for representing a single point (it's just a list of variables).
C. [tex]\((x, y, v)\)[/tex] - This uses the correct notation with parentheses, but the third coordinate should be [tex]\(z\)[/tex], not [tex]\(v\)[/tex].
D. [tex]\((w, x, y)\)[/tex] - This uses parentheses, but the variables indicate different axes since normally [tex]\(w\)[/tex] is not used as a coordinate in the standard three-dimensional Cartesian coordinate system.
From the analysis, the correct notation for representing a point in three-dimensional Cartesian coordinates is:
[tex]\[ \boxed{(x, y, z)} \][/tex]
Therefore, the correct answer is:
A. [tex]\((x, y, z)\)[/tex].
