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The rectangular coordinate system, often called the Cartesian coordinate system, is a method of defining the position of points in 2D or 3D space. It uses three axes, usually labeled as the X-axis, Y-axis, and Z-axis, which are all perpendicular to each other.
This setup allows us to measure distances and directions from a fixed origin, generally denoted as \(0,0,0\). Anytime you graph equations in three dimensions, you are effectively placing them within this rectangular framework. By plotting points on these axes, we can visualize complex geometric shapes in space.

• The X-axis refers to the horizontal line
• The Y-axis indicates the vertical line
• The Z-axis moves in and out of the plane

When dealing with three-dimensional shapes, using the coordinate system is essential to representing and understanding their spatial properties. It is the foundation for plotting points and visualizing shapes like spheres.

A quadratic equation is any equation that can be rearranged into the standard form ax^2 + bx + c = 0, where a, b, and c are constants. However, in the realm of three dimensions, quadratic equations often incorporate three variables, as seen in our exercise: x^2 + y^2 + z^2 - 4z = 0.
Quadratics in 3D can represent various shapes, such as spheres, ellipsoids, or paraboloids. Here, the equation signifies a sphere. The term "quadratic" indicates that each variable is squared, which is an essential feature of these equations.
Such equations enable us to analyze and solve complex geometric problems by manipulating and rearranging them into recognizable forms.

Completing the square is a method used to simplify quadratic expressions and equations. It involves transforming a quadratic equation into a perfect square trinomial, making it easier to solve or further manipulate. For our example, the equation x^2 + y^2 + z^2 - 4z = 0 can be difficult to interpret at first. However, by focusing on the z-term, we can simplify it. We rearranged it into x^2 + y^2 + (z^2 - 4z), allowing us to complete the square for z:

• Take half of the coefficient of z, which is -4, resulting in -2.
• Square -2 to get 4.
• Add and subtract this square inside the expression, yielding (z-2)^2 - 4.

This adjustment helps us transform the equation into a recognizable form, x^2 + y^2 + (z-2)^2 = 4, revealing crucial geometric information.

In 3D geometry, a sphere is a perfectly symmetrical object and can be described by a specific equation similar to that seen in our task. The general equation for a sphere in 3D is (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2. Here, \(h,k,l\) represents the center of the sphere and \(r\) is its radius.
In our problem, the transformed equation x^2 + y^2 + (z-2)^2 = 4 indicates a sphere centered at \(0,0,2\) with a radius of 2. Recognizing this form allows one to immediately identify and sketch the sphere in the rectangular coordinate system.

• The center is the point from which all points on the sphere are equidistant.
• The radius is the distance from the center to any point on the sphere's surface.

Understanding this can help visualize the sphere's structure and position it correctly in 3D space, based on the given equation.