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In three-dimensional space, the equation \( x^2 + y^2 + z^2 = r^2 \) represents a sphere with a center at the origin \((0, 0, 0)\) and a radius \( r \). This equation shows that every point \((x, y, z)\) on the surface of the sphere is at a constant distance, \( r \), from the center. The given exercise presents the equation \( x^2 + y^2 + z^2 = 81 \). To find the radius of the sphere, we equate \( r^2 \) to 81, leading to \( r = \sqrt{81} = 9 \). Thus, the sphere has a radius of 9 and is centered at the origin. This formula is very helpful since it allows us to easily identify the main characteristics of the sphere just by looking at the equation. Remember, the key parts of this equation are the variables \(x\), \(y\), and \(z\) squared, all summing to equal the square of the radius.

The three-dimensional coordinate system, often called three-space, is useful for visualizing objects like spheres. It has three axes: \(x\), \(y\), and \(z\). These axes are mutually perpendicular, allowing for the location of any point in space using three coordinates \((x, y, z)\).

• The origin, \((0, 0, 0)\), is the point where all three axes intersect.
• The positive direction is typically up or right along each axis, while the negative direction is down or left.
• Each coordinate represents a distance from the origin along one of the axes.

You can imagine the 3D space as a box where any point inside or on the surface of the box is designated by its \(x\), \(y\), and \(z\) values. This system makes it easier to determine the position and shape of 3D objects, like spheres, by allowing us to describe their geometry precisely.

Spheres have several distinct properties that are important in three-space. One of these is symmetry, as every point on the surface is equidistant from its center point. For a sphere centered at the origin with radius \( r \), all points \((x, y, z)\) that satisfy the equation \( x^2 + y^2 + z^2 = r^2 \) lie on the surface of the sphere. Some key properties include:

• Surface Area: The formula for the surface area of a sphere is \( 4\pi r^2 \).
• Volume: The volume is calculated using \( \frac{4}{3}\pi r^3 \).
• Symmetry: Spheres possess rotational symmetry around all axes through their center.

These attributes describe both the sphericity and the scales of the sphere, like how much space it encloses and how large its surface is. By understanding these properties, we gain deeper insights into the sphere's nature and its practical applications in fields like physics and engineering.