Problem 9 - Precalculus with Calculus Previews

Question

Describe geometrically all points \(P(x, y, z)\) whose coordinates satisfy the given conditions. $$ x=2, y=3 $$

Step-by-Step Solution

Answer

The points describe a line parallel to the z-axis at (x, y) = (2, 3).

Step 1: Understand the Conditions

We are given two equations that define the points, namely, \(x = 2\) and \(y = 3\). This means we're looking only for points where these conditions hold true, regardless of the value of \(z\).

Step 2: Interpret the Conditions in Three-Dimensional Space

Since \(x\) and \(y\) have fixed values (2 and 3 respectively), the description implies that the points \((x, y, z)\) are where the value of \(z\) can vary freely. This indicates a line parallel to the z-axis.

Step 3: Describe the Line Geometrically

The points satisfying \(x = 2\) and \(y = 3\) form a vertical line parallel to the \(z\)-axis in three-dimensional space. This line passes through the point \((2, 3, 0)\) on the \(xy\)-plane and extends infinitely in both positive and negative directions along the \(z\)-axis.

Key Concepts

Coordinates in 3D space

In the world of three-dimensional geometry, every point can be identified using a set of three coordinates:

• The x-coordinate tells us the position of the point in the direction of the x-axis.
• The y-coordinate defines how far the point is along the y-axis.
• The z-coordinate specifies the height of the point from the xy-plane along the z-axis.

By combining these three values, we can pinpoint any location within 3D space.
The concept is similar to locating a place on a map, but with the added detail of elevation or depth. For any set of coordinates \((x, y, z)\), each value has an impact on the position of the point.
In scenarios where one or two of these coordinates are fixed, such as \(x = 2\) and \(y = 3\), the points are constrained in a specific manner, leading to structures like lines or planes.

Lines parallel to an axis

When discussing lines in 3D space, lines parallel to an axis become important constructs.
If a line is described by keeping two coordinates constant and allowing the third to vary, this line will be parallel to the axis associated with that variable coordinate.

• For instance, with fixed coordinates \(x = 2\) and \(y = 3\), we have a scenario where the third coordinate \(z\) is free to change.

This yields a line parallel to the z-axis.
This line does not tilt or shift; it travels vertically up and down along the z-axis.
Lines parallel to an axis are pivotal in developing geometric and spatial understanding as they present linear paths crossing the planes, which are integral to a 3D coordinate system.

Geometric interpretation

Understanding the geometric significance of constant coordinate conditions enriches our comprehension of spatial layouts.
For the condition \(x = 2\) and \(y = 3\), we can visualize a line running parallel to the z-axis. This line starts at the point \((2, 3, 0)\) in the xy-plane and shoots both upwards and downwards along the z-axis indefinitely.
Such a line doesn't occupy an area or volume, but it defines a direction and spatial relation, serving as a foundational building block in 3D geometry.

• The point \((2, 3, 0)\) can be viewed as the anchor or starting point on the xy-plane where this line intersects before continuing along the z-axis.

Conceptualizing this helps not only in theoretical mathematics but also in practical fields like engineering, where understanding spatial extensions and projections is crucial.
Thus, the geometric interpretation aids in identifying and visualizing how specific coordinate constraints manifest within an infinite three-dimensional canvas.