Problem 18 - Advanced Engineering Mathematics

Question

Describe the locus of points \(P(x, y, z)\) that satisfy the given equation(s). $$ (x-2)(z-8)=0 $$

Step-by-Step Solution

Answer

The locus is the union of the planes \(x = 2\) and \(z = 8\).

Step 1: Understand the Equation

Recognize that the equation \((x-2)(z-8) = 0\) defines conditions for when the product of two terms is zero. For a product to be zero, at least one of the individual terms must be zero.

Step 2: Break Down the Equation

Break down the equation \((x-2)(z-8) = 0\) into two separate cases: \(x-2 = 0\) and \(z-8 = 0\).

Step 3: Solve Case 1

For the first case \(x-2 = 0\), solve for \(x\). This gives: \(x = 2\). This represents a plane where all points have the same x-coordinate of 2, irrespective of y and z.

Step 4: Solve Case 2

For the second case \(z-8 = 0\), solve for \(z\). This gives: \(z = 8\). This represents a plane where all points have the same z-coordinate of 8, irrespective of x and y.

Step 5: Combine the Solutions

The locus of points that satisfy the original equation is the union of the two solutions. Thus, the locus is the set of all points on the plane \(x = 2\) together with the set of all points on the plane \(z = 8\). These two planes intersect at the line where \(x = 2\) and \(z = 8\).

Key Concepts

Three-Dimensional Geometry

Three-dimensional geometry extends our familiar two-dimensional view into a new dimension, adding depth to the concepts we know. In 3D, every point is described by three coordinates: \((x, y, z)\), which represent its position along the x, y, and z axes, respectively.
Unlike in 2D, where only x and y determine a point's position, having a third axis allows for describing objects like cubes, spheres, and planes in much richer detail.

• The x-axis usually represents the horizontal position.
• The y-axis is vertical in most representations.
• The z-axis adds depth, sticking out of the plane defined by x and y.

This 3D view is crucial in many fields, from engineering to animation, helping us conceive objects as they exist in the real world.

Plane in 3D Space

A plane in three-dimensional space is a flat, two-dimensional surface that extends infinitely in all directions within its dimension. Planes can slice through the 3D space, creating sections at various angles and positions. Typically, a plane is described using a linear equation of the form \(Ax + By + Cz + D = 0\). This equation specifies a condition that points on the plane must satisfy.
In the given exercise, we see a simpler expression separately for each plane described as \(x = 2\) and \(z = 8\).

• When \(x = 2\), you're defining a vertical plane parallel to the yz-plane, shifted along the x-axis.
• When \(z = 8\), it forms a horizontal plane parallel to the xy-plane, elevated along the z-axis.

These planes create interfaces, sections, and boundaries in the 3D space, useful for modeling and analysis.

Intersection of Planes

When two planes intersect in 3D space, they meet along a line. This is a key concept in 3D geometry as it simplifies complex spatial relationships, often reducing a problem to a one-dimensional form — a line.
In the context of the given exercise, the equation

• \(x = 2\)
• \(z = 8\)

Describes two separate planes that intersect at the line determined by these constant values.
This intersection results in the line where both planes share a common set of points; any point on this line will have \(x = 2\) and \(z = 8\). This simplifies visualization and calculation because instead of dealing with a plane's infinite extent, we focus on a single line of intersection.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra with geometry to describe geometry in numerical terms. By using coordinate systems, every geometric element can be represented with numbers, making calculations straightforward.
In this approach, planes and lines in 3D space are represented as equations, as we've seen with \(x = 2\) and \(z = 8\). These are not just abstract concepts; they are concrete algorithms for describing spatial properties and relationships.

• You can solve geometric problems using algebraic equations.
• Coordinate paths simplify the understanding of intersections, distances, and angles.

Coordinate geometry enables students to transition from visualizing objects in space to performing calculations that model these very objects, enhancing both theoretical understanding and practical problem-solving skills.