Answered

If you use a pair of [tex]$x$[/tex]- and [tex]$y$[/tex]-coordinates [tex]$(x, y)$[/tex] to represent a point in a two-dimensional Cartesian coordinate system, how would you represent a point in a three-dimensional Cartesian coordinate system?

A. [tex]$(x, y, z)$[/tex]
B. [tex]$x, y, z$[/tex]
C. [tex]$(x, y, v)$[/tex]
D. [tex]$(w, x, y)$[/tex]



Answer :

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].