Question

What is the \(z\) -coordinate of any point in the \(x y\) -plane?

Short answer

The z-coordinate of any point in the x-y plane is always zero.

Step-by-step solution

Understanding the 3D Coordinate System

In a three-dimensional coordinate system (which includes the 'x' axis, 'y' axis and 'z' axis), every point in space is described by three coordinates: chosen distances from three mutually orthogonal (i.e., at right angles to each other) lines known as the x-axis, y-axis, and z-axis. The intersection of these axes,, forms the origin (0,0,0).

Identifying the plane

The x-y plane is defined as the planar sector in the 3D space where the 'z'-coordinate of any point on this plane is always zero, regardless of the values of 'x' and 'y'.

Final Statement

Therefore, the z-coordinate of any point lying on the x-y plane in three dimensions is always zero.

Key concepts

Z-Coordinate in 3D Space

The z-coordinate is a fundamental component within the three-dimensional (3D) coordinate system, allowing us to locate any point along the vertical axis. In essence, the z-coordinate is what separates the 3D space from its 2D counterpart by adding depth to the horizontal plane formed by the x and y axes. It's like a building's floor number—while the address (x,y) gives you the position on the map, the z-coordinate tells you how high up you need to go.

Imagine you are in an elevator, and each button represents a different z-coordinate. If someone asks for the z-coordinate on the ground floor (the xy-plane), you'd confidently press the button labeled '0' because, on this level, no matter where you stand—be it the lobby or right in front of the café—the height is the same. This scenario mirrors the solution stating that any point on the xy-plane will always have a z-coordinate of zero.

XY-Plane

The xy-plane is a flat, two-dimensional surface that stretches infinitely in 3D space. Think of it as a vast sheet of paper that extends without end in all the horizontal directions—you can draw or plot any point using only the x and y values since the z-value will be consistent... precisely zero, akin to the paper lying flat on a table. It's much like playing a classic video game where the characters can only move left, right, up or down but not up into the air or down into the ground.

When we say a point lies on the xy-plane, we imply that it's pinned down to that flat surface. It's an important concept because the xy-plane serves as a reference frame; it gives a standardized base for describing the position of any point above or below it in the third dimension.

Orthogonal Axes

The term 'orthogonal' stems from Greek origins, meaning 'right angle,' and in the context of a 3D coordinate system, it refers to the fact that the x, y, and z axes are all perpendicular to each other.

Picture a room’s corner where two walls meet the floor, each forming a 90-degree angle. This meeting point is much like the three orthogonal axes coming together at the origin. These axes allow us to specify any location in space with precision. Every axis serves as a ruler, measuring distances from the origin in their respective directions: horizontally, vertically, and laterally.

If axes were not orthogonal, measuring would become a tricky task, like trying to follow directions based on skewed or slanted signposts. But with orthogonality, we have a reliable, straightforward, and universally understood system that navigates us through the complexities of 3D space.