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Chapter 7: Problem 11

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

Short Answer

Expert verified
The \(z\)-coordinate of any point in the \(x\)-plane is 0.

Step by step solution

01

Understanding the 3D Cartesian coordinate system

In a 3D Cartesian coordinate system, any point is represented as \((x, y, z)\) where \(x\), \(y\), and \(z\) are the coordinates along the x, y, and z axes respectively.
02

Identifying the x-plane

The \(x\)-plane is a 3-dimensional plane where \(z = 0\). This is regardless of the values of \(x\) and \(y\).
03

Recognize the value of z-coordinate

Since any point on the \(x\)-plane has \(z = 0\), the \(z\)-coordinate of any point on the \(x\)-plane is 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

x-plane
In the 3D Cartesian coordinate system, the concept of an "x-plane" might sound a bit confusing at first. Imagine a flat surface that extends infinitely in the 3D space. This plane is defined by setting the
  • z-coordinate to 0
This means any point you pick on this particular plane will always have its
  • z-coordinate as 0
Think of the
  • x-plane as the equivalent of a 2D coordinate plane embedded within this 3D space
No matter how high or low you go in the directions of the x or y axes, nothing changes the
  • z-coordinate
To better visualize, think of a horizontal table top where the table's surface represents the
  • x-plane
    • In summary, every point on the x-plane in a 3D Cartesian system shares this common feature: its z-coordinate remains steadfastly at zero.
    z-coordinate
    The z-coordinate is an essential part of the 3D Cartesian coordinate system because it helps specify exactly where a point stands in the three-dimensional space. In a 3D system, every point is expressed as
    • extbf{(x, y, z)}
    where
    • z
    refers to the point's position along the z-axis. This can be thought of as the depth or elevation from the reference plane, usually the 'ground' when drawing parallels from our natural view of the world.
    • A zero value for the z-coordinate signifies that the point lies directly on the x-plane
    • A positive z-coordinate suggests height or elevation above the plane
    • A negative z-coordinate indicates the point is beneath the base plane level
    Understanding the z-coordinate, therefore, provides critical insight into the topography of the space you are analyzing.
    Cartesian coordinates
    In mathematics and geometry, the Cartesian coordinate system is a universal framework that allows us to pinpoint the precise location of a point within a defined space. In 2D, we have certainly encountered it before as a simple grid of x and y-axes. When extended into three dimensions, we introduce the
    • z-axis, adding depth to our understanding of space
    In this 3D Cartesian space, every location can be specified by an ordered triplet
    • (x, y, z)
    • x-coordinate: represents the horizontal position
    • y-coordinate: denotes vertical positioning
    • z-coordinate: adds the dimension of depth or height
    The 3D Cartesian coordinate system is particularly powerful because it gives us a clear, mathematical method for describing 3-dimensional space. By understanding how these coordinates interact, one can accurately map and navigate through any conceivable 3D environment.

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