In order to represent a point in a three-dimensional Cartesian coordinate system, you need to include three coordinates, typically corresponding to the x, y, and z axes. Here’s a step-by-step explanation of what each option represents and why option D is correct:
1. Understanding the notation in two dimensions:
- In a two-dimensional Cartesian coordinate system, a point is represented using two coordinates, usually denoted as [tex]\((x, y)\)[/tex].
- The [tex]\(x\)[/tex]-coordinate indicates the position along the horizontal axis.
- The [tex]\(y\)[/tex]-coordinate indicates the position along the vertical axis.
2. Introducing the third dimension:
- In a three-dimensional Cartesian coordinate system, we need to represent a point with three coordinates.
- Besides the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates, the additional coordinate, typically denoted as [tex]\(z\)[/tex], represents the position along the depth axis (perpendicular to both the x and y-axes).
3. Evaluating the given options:
- Option A: [tex]\((x, y, v)\)[/tex]: This uses [tex]\(v\)[/tex] instead of [tex]\(z\)[/tex]. It is not the conventional way to represent a three-dimensional point.
- Option B: [tex]\((w, x, y)\)[/tex]: This option uses [tex]\(w\)[/tex] instead of [tex]\(z\)[/tex] and rearranges the usual order. It's also unconventional.
- Option C: [tex]\(x, y, z\)[/tex]: This seems close but does not use the parentheses to clearly delineate the coordinates of a point.
- Option D: [tex]\((x, y, z)\)[/tex]: This matches the conventional notation, where each coordinate is enclosed within parentheses.
Therefore, the proper way to represent a point in a three-dimensional Cartesian coordinate system is:
[tex]\[ \boxed{(x, y, z)} \][/tex]
Hence, the correct answer is:
[tex]\[ \text{Option D. } (x, y, z) \][/tex]
