Problem 15 - Advanced Engineering Mathematics

Question

Describe the locus of points \(P(x, y, z)\) that satisfy the given equation(s). $$ x y z=0 $$

Step-by-Step Solution

Answer

The locus is the union of the xy-plane, yz-plane, and xz-plane.

Step 1: Understanding the equation

The equation \(xyz = 0\) signifies that at least one of the coordinates \(x\), \(y\), or \(z\) must be zero. This equation provides three cases to consider: \(x = 0\), \(y = 0\), or \(z = 0\).

Step 2: Case 1: x = 0

If \(x = 0\), points lie on the yz-plane. This plane includes all points where the x-coordinate is zero but y and z can be any real numbers. The equation simplifies to \(0 \times y \times z = 0\), which is always true regardless of the values of \(y\) and \(z\).

Step 3: Case 2: y = 0

If \(y = 0\), points lie on the xz-plane. This plane includes all points where the y-coordinate is zero but x and z can be any real numbers. The equation simplifies to \(x \times 0 \times z = 0\), which is always true regardless of the values of \(x\) and \(z\).

Step 4: Case 3: z = 0

If \(z = 0\), points lie on the xy-plane. This plane includes all points where the z-coordinate is zero but x and y can be any real numbers. The equation simplifies to \(x \times y \times 0 = 0\), which is always true regardless of the values of \(x\) and \(y\).

Step 5: Combining all cases

The union of the three planes reconfirms the solution to the equation \(xyz = 0\). Any point \(P(x, y, z)\) that lies on at least one of these three planes satisfies the given equation: \(x = 0\), \(y = 0\), or \(z = 0\). Therefore, the locus of points that satisfy the equation consists of the union of the yz-plane, xz-plane, and xy-plane.

Key Concepts

Coordinates

In three-dimensional space, the position of any point is determined using a set of numbers called coordinates. These coordinates are denoted as

• \(x\),
• \(y\), and
• \(z\).

Each number corresponds to a location along one of the three axes: the x-axis, y-axis, and z-axis, respectively.
Coordinate systems are essential because they provide a structured way to describe the location of points, lines, and entire spaces within 3D geometry.
In our particular exercise, we explore points \( P(x, y, z) \) that satisfy a specific condition within this coordinate system.

3D Planes

A plane in 3D space can be thought of as a flat, two-dimensional surface extending indefinitely in all directions within its dimensions. In this context, planes can be defined using a coordinate equation.
For example, the xz-plane is defined as the set of all points where \(y = 0\).
This means that every point on this plane will have a y-coordinate of zero, while the x and z coordinates can take any value. Similarly,

• the yz-plane is defined by \(x = 0\), and
• the xy-plane by \(z = 0\).

Planes are an integral part of geometry as they help understand the spatial relationships between different objects and their positions.

Equation Satisfying Points

When we talk about points satisfying an equation like \(xyz = 0\), this refers to identifying all combinations of coordinates \((x, y, z)\) that make the equation true.
In this case, at least one of the coordinates must be zero for the equation to hold.
This creates three possibilities or cases based on which coordinate is zero:

• \(x = 0\) (points on the yz-plane),
• \(y = 0\) (points on the xz-plane),
• \(z = 0\) (points on the xy-plane).

These planes intersect at the axes lines, adding another layer to the solution.
Thus, the locus or the set of all points satisfying the equation is the union of these three planes, demonstrating how algebraic equations can represent geometric objects.