Question

When stating the coordinates of a point, which coordinate is given first?

Short answer

Answer: The x-coordinate is given first when stating the coordinates of a point in both 2D and 3D coordinate systems.

Step-by-step solution

Identify the Coordinate System

Before identifying which coordinate is given first, we must decide whether the point is in a two-dimensional (2D) or a three-dimensional (3D) coordinate system. In a 2D coordinate system, we have two coordinates (x and y), while in a 3D coordinate system, we have three coordinates (x, y, and z).

The Order of Coordinates in a 2D Coordinate System

In a 2D coordinate system, a point is represented by the coordinates (x, y). Here, the x-coordinate is given first, followed by the y-coordinate. For example, if you have coordinates (3, 2), the x-coordinate is 3, and the y-coordinate is 2.

The Order of Coordinates in a 3D Coordinate System

In a 3D coordinate system, a point is represented by the coordinates (x, y, z). Similar to the 2D coordinate system, the x-coordinate is given first, followed by the y-coordinate and the z-coordinate. For example, if you have coordinates (4, -1, 6), the x-coordinate is 4, the y-coordinate is -1, and the z-coordinate is 6.

Conclusion

In both 2D and 3D coordinate systems, the x-coordinate is given first when stating the coordinates of a point. This is the standard convention adopted in mathematics and physics.

Key concepts

2D Coordinates

In the realm of mathematics, understanding 2D coordinates is fundamental. Imagine a flat piece of paper; this represents a two-dimensional space. On this plane, every point can be identified using a pair of numbers known as coordinates. These are often referred to as the x-coordinate and y-coordinate.

The arrangement follows a particular order: the x-coordinate comes first, followed by the y-coordinate, as in (x, y).

• The x-coordinate represents the horizontal position, indicating how far left or right the point is.
• The y-coordinate denotes the vertical position, showing how far the point is up or down.

Visualize a graph where the x-axis is the horizontal line and the y-axis is the vertical line. Any point on the graph can be pinpointed using these coordinates. For instance, the point (3, 2) would mean moving 3 units right and 2 units up from the origin (0, 0).

This two-number system is essential for plotting graphs, navigating maps, and even designing graphics in computer programs.

3D Coordinates

When you step into the world of 3D coordinates, you're venturing into the three-dimensional space, much like our day-to-day world. In this space, a point is determined not by two, but three numbers known as coordinates: x, y, and z.

The notation follows (x, y, z), where each component serves a unique purpose in defining position:

• x-coordinate: This signifies the position along the horizontal axis, similar to the 2D system.
• y-coordinate: This represents the vertical axis, again akin to the 2D setup.
• z-coordinate: This is where the additional dimension comes in, showing the depth or height, indicating how far in or out the point is from the viewer.

Imagine a room where the floor is your x-y plane, and the height up to the ceiling represents the z-axis. To locate a point such as (4, -1, 6), move 4 units along the x-axis, -1 along the y-axis, and then either up or down 6 units along the z-axis. Understanding 3D coordinates is crucial not just in math, but also in fields like physics, architecture, and computer graphics, allowing us to model and interact with three-dimensional objects.

Cartesian Coordinates

The Cartesian coordinate system, named after the French mathematician René Descartes, is the backbone of both 2D and 3D systems. This system employs a perpendicular grid of axes that intersect at a point called the origin, typically marked as (0, 0) in 2D and (0, 0, 0) in 3D.

What makes the Cartesian system powerful is its simplicity and consistency across dimensions:

• In 2D, it consists of two intersecting lines: the x-axis and the y-axis.
• In 3D, a third line, the z-axis, is added, forming a three-way intersection at the origin.

Each point's position is described by its distance from each of these axes, following the standard sequencing of x, y, and potentially z for 3D. A Cartesian plane helps in visualizing algebraic equations, tracking geometric shapes, or even conducting data analysis. By plotting points, lines, and curves, one can gain insights into the relationships and functions that govern mathematical and physical realities.

Conclusively, whether you're dealing with 2D or 3D, Cartesian coordinates offer a universal language for discussing positions and movements within math's vast landscape.