Problem 9 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

The points form a vertical line in 3D space at \( (x, y) = (2, 3) \).

Step 1: Understand the Conditions

We need to understand the given conditions. The conditions specified are \( x = 2 \) and \( y = 3 \). This means any point \( P(x, y, z) \) must have its \( x \)-coordinate at 2 and its \( y \)-coordinate at 3.

Step 2: Interpret the Conditions Geometrically

The conditions \( x = 2 \) and \( y = 3 \) restrict movement in the \( x \)- and \( y \)-directions. Geometrically, this means that all points satisfying these conditions lie in a plane parallel to the \( xy \)-plane, but specifically at the line where \( x = 2 \) and \( y = 3 \) throughout the \( z \)-direction.

Step 3: Describe the Locus of Points

Since only the \( z \)-coordinate is free to vary, the locus of points is a vertical line in 3D space. This line travels through all potential values of \( z \), while \( x \) and \( y \) are fixed at 2 and 3, respectively.

Key Concepts

Coordinate System

A coordinate system is like a map for three-dimensional space. In 3D, we usually use three axes: the x-axis, y-axis, and z-axis.
Each point in this space is represented by a set of three values or coordinates - one for each axis. For example, a point \( P(x, y, z) \) has coordinates that tell you how far it is along the x-axis, how far along the y-axis, and how far up or down it is along the z-axis.

• The x-coordinate shows how far along the x-axis you go.
• The y-coordinate shows how far along the y-axis you go.
• The z-coordinate indicates height or depth relative to the base plane.

Understanding these coordinates helps visualize points in 3D space, like placing a point on a 3D graph. Everything unfolds from these three values, including movements and positions.

Plane

In 3D geometry, a plane is like a flat sheet that stretches infinitely in two directions. It has no thickness, just like a piece of paper without top or bottom, extending endlessly through space.
When we say the conditions \( x = 2 \) and \( y = 3 \) specify a plane, we mean we're looking at all points where these conditions are true.
In this case, such a plane is parallel to the z-axis.

• This plane will hold all points where \( x \) = 2 and \( y \) = 3 for any value of \( z \).
• Unlike a typical plane across the x and y axes, this one runs vertically and endlessly in the z-direction.

Visualizing this, imagine holding a sheet of glass upright at the position x = 2 and y = 3. It shows all the locations possible on the z-axis with fixed x and y.

Locus of Points

The locus of points is a set of points satisfying specific conditions.
When only the z-coordinate is variable, while x and y are fixed, the locus becomes a straight line across the entire z-axis. It's like drawing a vertical line from the floor to the ceiling, where you can only move up or down, not side to side.
In our example, with \( x = 2 \) and \( y = 3 \), the possible points create a locus that is a vertical line.

• The line extends infinitely in the positive and negative z-directions.
• It creates a simple path through all possible z-values, marking a distinct trace in 3D space.

This understanding helps to see how points behave spatially, forming lines, curves, or complex shapes based on their defining conditions.