Problem 8 - Precalculus with Calculus Previews

Question

Describe geometrically all points \(P(x, y, z)\) whose coordinates satisfy the given conditions. $$ z=5 $$

Step-by-Step Solution

Answer

The points form a plane parallel to the xy-plane at \( z = 5 \).

Step 1: Understand the equation

The equation given is \( z = 5 \). This equation specifies a condition for the z-coordinate of any point \( P(x, y, z) \).

Step 2: Interpret the z-coordinate condition geometrically

Since the condition \( z = 5 \) is applied to the z-coordinate only, it implies that every point \( P(x, y, z) \) must have its z-coordinate exactly equal to 5.

Step 3: Identify the geometric object defined by the condition

The condition \( z = 5 \) does not place any restrictions on the x or y coordinates, meaning they can take any real value. This results in a plane parallel to the xy-plane at z = 5.

Key Concepts

Cartesian Planes

Cartesian planes are a fundamental concept in 3D geometry, building upon the familiar 2D planes.
In a 3-dimensional space, the Cartesian plane system is an extension of the xyz coordinate system.
Here's how it breaks down:

• The x-coordinate measures horizontal movement, left or right.
• The y-coordinate measures vertical movement, up or down.
• The z-coordinate measures depth, how far "in" or "out" a point is from a reference point, known as the origin (0,0,0).

When you set a condition like \( z = 5 \), it means that every point satisfying this condition is positioned on a plane where every point's z-value is 5.
This plane is parallel to the xy-plane (where z is usually 0) but shifted "up" 5 units in the space where z increases.

Geometric Interpretation

Geometrically understanding dimensions in 3D is crucial in coordinate geometry.
When we say \( z = 5 \), we are visualizing a specific location for all points sharing this z-value.
The condition implies that:

• Every point lies on a flat surface (a plane).
• This flat surface is horizontal, like a tabletop.
• The surface is located at z = 5, evenly distributing every point 5 units above the xy-plane.

Restrictions-free x and y coordinates mean these points can scatter anywhere sideways across this plane.
Thus, the collection of points forms an infinite flat sheet parallel to the ground.

Spatial Analysis

Spatial analysis involves evaluating how different geometric objects interact in space.
A plane defined by \( z = 5 \) highlights how a fixed position in one axis allows free movement in the other two.
By being parallel to the ground (xy-plane), this specific plane:

• Separates space into distinct regions above and below itself.
• Is an intersection of 3D space plans, showing where specific conditions apply.
• Enables movement along x and y without constraint, showcasing the dimensional flexibility of surfaces.

Understanding spatial relationships like this helps solve practical problems, placing points, or even visualizing how objects exist and traverse within the space.