Problem 93 - Precalculus with Calculus Previews

Question

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

Step-by-Step Solution

Answer

Points on or outside a sphere with radius 1 centered at the origin.

Step 1: Identify the Equation

The equation given is \( x^2 + y^2 + z^2 \geq 1 \). This represents a condition involving three variables \(x\), \(y\), and \(z\), which are the coordinates in 3-space.

Step 2: Recognize the Shape

The equation \( x^2 + y^2 + z^2 = 1 \) defines the surface of a sphere centered at the origin with radius 1. Therefore, the inequality \( x^2 + y^2 + z^2 \geq 1 \) involves points that are on or outside this sphere.

Step 3: Describe the Region Geometrically

All points whose coordinates satisfy \( x^2 + y^2 + z^2 \geq 1 \) form the region outside and inclusive of a sphere with radius 1. This means the set is the complement of the open ball of radius 1 centered at the origin, including the boundary points.

Key Concepts

Understanding Spheres in 3-Space

In the world of geometry, a sphere is a perfectly symmetrical 3-dimensional shape that is defined in 3-space. It is the set of all points that are at a given distance from a central point, known as the center. This given distance is the radius of the sphere. In mathematical terms, if a sphere is centered at a point \(a, b, c\), then any point \(x, y, z\) on the surface of the sphere satisfies the equation: \[ (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2 \] - Here, \(r\) represents the radius of the sphere. - If the center is at the origin (0, 0, 0), the equation simplifies to \(x^2 + y^2 + z^2 = r^2\).
When the points satisfy an inequality such as \(x^2 + y^2 + z^2 \geq r^2\), it means they lie on the surface or outside this sphere. This forms a solid region extending infinitely in all directions outside the surface of the sphere. Such inequalities are key in understanding geometric boundaries in space, as they help define regions like the exterior of a sphere.

Interpreting Inequalities in Geometry

Inequalities in geometry help to understand regions and boundaries within 3-space. Specifically, an inequality like \(x^2 + y^2 + z^2 \geq 1\) indicates a region in space instead of a strict surface or line.
- This particular inequality encompasses all points for which the squared distances from the origin sum up to at least 1. - It includes both \(x^2 + y^2 + z^2 = 1\) (the boundary) and \(x^2 + y^2 + z^2 > 1\) (the space outside the sphere).
When interpreting these inequalities, you are essentially looking at a 3-dimensional shape and defining a space that contains points both on and beyond this shape. This is crucial in mathematical problems that involve constraints or conditions on the space where solutions or paths exist.

The 3-Space Coordinate System

The 3-space coordinate system extends the familiar two-dimensional Cartesian plane into three dimensions. This involves three coordinates \(x, y, \text{ and } z\). Each coordinate corresponds to a point's alignment along one of the three axes:

• The \(x\)-axis runs horizontally.
• The \(y\)-axis runs vertically (in a traditional 2D view).
• The \(z\)-axis adds depth, extending perpendicularly out of the xy-plane.

This 3D system allows us to locate any point in space unequivocally. The point \( (x, y, z) \) is a unique position where these coordinates intersect. Understanding this system is crucial because geometrical shapes and inequalities use these coordinates to define areas, surfaces, or volumes in space. Grasping how objects like spheres and other shapes are expressed helps students in visualizing and solving more complex spatial problems.