Problem 94 - Precalculus with Calculus Previews

Question

Describe geometrically all points in 3-space whose coordinates satisfy the given condition(s). $$ 0<(x-1)^{2}+(y-2)^{2}+(z-3)^{2}<1 $$

Step-by-Step Solution

Answer

The points form an open sphere centered at (1, 2, 3) with radius 1, excluding the center.

Step 1: Understand the Inequality

The inequality \(0 < (x-1)^2 + (y-2)^2 + (z-3)^2 < 1\) represents all points in 3D space such that the distance from the point \((x, y, z)\) to the center \((1, 2, 3)\) is greater than \(0\) and less than \(1\). This excludes the point being exactly at the center.

Step 2: Recognize the Geometric Shape

The given inequality describes a sphere centered at the point \((1, 2, 3)\) with a radius slightly less than \(1\). Points on the boundary at \((x-1)^2 + (y-2)^2 + (z-3)^2 = 1\) form a sphere. The condition \((x-1)^2 + (y-2)^2 + (z-3)^2 > 0\) indicates that points don't include the center.

Step 3: Combine Conditions

Putting both conditions together, the inequality represents all points inside the sphere centered at \((1, 2, 3)\) with radius \(1\), excluding the center itself. This can be visualized as an open sphere, with no points on or inside the circle of radius \(0\).

Key Concepts

Inequalities in three dimensions

When we enter into the realm of 3D geometry, inequalities help us understand how to express volumes and regions in three-dimensional space. The inequality \(0 < (x-1)^2 + (y-2)^2 + (z-3)^2 < 1\) is an excellent example.This inequality indicates a special condition:- The sum must be greater than zero, which means points cannot touch the coordinate at the center \((1, 2, 3)\).- Meanwhile, it must remain less than one, meaning points cannot lie on the boundary of the region that forms a sphere.Inequalities like this are crucial for defining regions with specific spatial properties, allowing visualization of complex structures within coordinate systems.

Sphere in coordinate space

In coordinate geometry, a sphere emerges as a crucial concept. It's defined mathematically by addressing the distance of all points from a central point in space, which is known as the sphere's center. The spherical inequality \((x-1)^2 + (y-2)^2 + (z-3)^2 = 1\) represents a sphere with a fixed radius of 1 unit centered at \((1, 2, 3)\).However, within our stated inequality, we modify the properties slightly:- The region is formed by excluding points exactly 1 unit away (resulting from the \(< 1\) portion), making it an **open sphere**.- The inequality \(> 0\) ensures no point is exactly at the center.This sphere doesn't include its surface and certainly not the center, leading to interesting explorations in 3D spaces and models.

Geometric interpretation of inequalities

Geometrically interpreting inequalities involves visualizing how they shape regions in space. The inequality in question depicts a geometric portrayal of points nestled within a spherical shape, but still not on its surface or its center.Key aspects include:- **Open Sphere:** The inequality creates a space that can be thought of as a sphere, where the boundaries are strictly defined. The sphere is open, meaning the boundary isn't included.- **Exclusionary Center:** The zero point, or the center at \((1, 2, 3)\), is excluded. No point coincides with this spot.This interpretation assists in visualizing mathematical concepts, providing physical representations for abstract mathematical conditions.