Problem 10 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

The condition describes a single point in 3D space at \((4, -1, 7)\).

Step 1: Identify the Fixed Coordinates

The condition provided is that the coordinates are fixed as \(x=4\), \(y=-1\), and \(z=7\).

Step 2: Understand the Meaning of Fixed Coordinates in Geometry

If all three coordinates \((x, y, z)\) of a point are fixed, it specifies a unique point in three-dimensional space.

Step 3: Describe the Geometrical Shape

In three-dimensional geometry, a specific point defined by fixed coordinates \((x=4, y=-1, z=7)\) represents a single point in the 3D space. There are no additional movements or dimensions involved. It does not form a line, plane, or any other shape.

Key Concepts

Fixed Coordinates

In the world of 3D geometry, the term "fixed coordinates" refers to a specific and unchanging set of values for a point in space. These coordinates do not change under any conditions. They are immutable values defining the precise location of a point. For instance, when we say a point has fixed coordinates \(x=4\), \(y=-1\), and \(z=7\), it means that no matter what, those values remain constant.
This concept is crucial in understanding geometric descriptions because:

• Each fixed coordinate corresponds to a specific axis in 3D space: \(x\) for the x-axis, \(y\) for the y-axis, and \(z\) for the z-axis.
• Changes in any of these coordinates signify a movement in the associated dimension, which is not the case when they are fixed.

Fixed coordinates help us pinpoint exact locations in three-dimensional space, making them fundamental to various applications such as computer graphics, robotics, and navigation systems.

Point in Space

A point in space is an exact location within a three-dimensional environment, identified by coordinates. In mathematical terms, a point \(P(x, y, z)\) does not have any size, shape, or dimension. It's merely a position.
In the context of the given problem, the point \(P(4, -1, 7)\) is defined by:

• The x-coordinate \(x=4\), which indicates the position on the x-axis.
• The y-coordinate \(y=-1\), which marks the location on the y-axis.
• The z-coordinate \(z=7\), which specifies the height on the z-axis.

Because a point has no dimensions, it can't "form" anything like a line or a surface. It's simply a mark of location. Thus, when these coordinates are fixed, as in our example, the point is permanently set in that particular space without any extension or spread.

Three-Dimensional Space

Three-dimensional space (3D space) is a geometric model of the physical universe in which we live, extending in three directions. Each point in this space can be described using three coordinates, each corresponding to one of the dimensions.The characteristics of 3D space include:

• The x-axis, which represents width or horizontal positioning.
• The y-axis, which signifies depth or vertical placement.
• The z-axis, which stands for height or elevation.

This coordinate system enables us to describe and locate any point within the spatial environment accurately. In a visual representation, imagine the coordinates \(x=4\), \(y=-1\), and \(z=7\). These form a single, unique point in this three-dimensional framework. It's akin to a GPS pin dropped at a precise intersection of width, depth, and height in our universe. Three-dimensional space is vital for understanding and navigating our world, encompassing everything from architecture and engineering to video games and virtual reality.