Problem 11 - Calculus an Applied Approach

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: Identifying the x-plane

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

Step 3: 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.

Key Concepts

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.