Problem 10 - Precalculus with Calculus Previews

Question

Describe geometrically all points \(P(x, y, z)\) whose coordinates satisfy the given conditions. $$ x=4, y=-1, z=7 $$

Step-by-Step Solution

Answer

The coordinates describe a single point: \((4, -1, 7)\).

Step 1: Understand the Given Conditions

We are given the coordinates of a point in three-dimensional space: \(x=4\), \(y=-1\), and \(z=7\). These coordinates are specific numerical values for the variables \(x\), \(y\), and \(z\).

Step 2: Geometric Interpretation of the Point

In the three-dimensional coordinate system, each point is identified by a triplet of numbers \((x, y, z)\). These numbers represent the position of the point along the x-axis, y-axis, and z-axis respectively.

Step 3: Locate the Point in Three-Dimensional Space

The given conditions specify a single point in space: \((4, -1, 7)\). This point is fixed at \(x = 4\), \(y = -1\), and \(z = 7\). It does not represent a line, plane, or any other geometric figure, but rather a single position in space.

Key Concepts

Geometric Interpretation

When talking about **geometric interpretation** in three-dimensional coordinates, we're dealing specifically with understanding points in three-dimensional space. Each point is defined by a set of three coordinates:

• **The x-coordinate** determines the position along the x-axis.
• **The y-coordinate** sets the position along the y-axis.
• **The z-coordinate** specifies the position along the z-axis.

In this system, a point is represented as \( P(x, y, z) \). For the problem given, we have the point \( P(4, -1, 7) \), which is located exactly where \( x = 4 \), \( y = -1 \), and \( z = 7 \).
Imagine a three-dimensional scatterplot. This representation helps you visualize the exact position of the point. Unlike in two dimensions, where it would sit on a flat surface, in three dimensions, it feels like plotting in a volume, allowing for a more complex spatial understanding.

Coordinate System

The **coordinate system** is fundamental for accurately locating positions in three-dimensional space. This system uses three perpendicular axes:

• The **x-axis** is usually drawn horizontally and represents width.
• The **y-axis** is often depicted vertically and indicates height.
• The **z-axis** extends front to back, showcasing depth.

These axes intersect at the origin point \( (0, 0, 0) \), creating quadrants that navigate the entire space surrounding it. Distances along each of these axes can be either positive or negative, revealing the point's direction from the origin.
In essence, think of it as a grid spreading throughout space, with coordinates like \( (4, -1, 7) \) letting us pinpoint a precise location. This grid-like system provides a structured way to map and communicate positions in three dimensions effectively.

Locating Points in Space

**Locating points in space** within this coordinate system is akin to finding a specific address in a vast neighborhood of locations. The coordinates given, \( (4, -1, 7) \), act like directions. You start at the origin and move:

• **4 units along the x-axis**
• **-1 unit along the y-axis**
• **7 units along the z-axis**

By following these steps, you find the exact point within the three-dimensional field. Visualizing this process helps to comprehend not just the position but also how the space is navigated - a crucial skill for any spatial reasoning task.
Triangles, lines, and surfaces are more freely formed here, but often they involve multiple points. However, points like \( (4, -1, 7) \) are unique to themselves, marking a solitary position in the vast emptiness or fullness of three-dimensional space.