Answer:
A Brief Introduction to
The set of all real numbers is denoted by . It is convenient to associate real numbers with points on a line, called the real number line.
The set of all ordered pairs , where and are real numbers, is called . Using set notation we write: Geometrically speaking, can be associated with a coordinate plane in which we refer to each point by its and coordinates.
The set of all ordered triples , where , and are real numbers, is called . Geometrically, an ordered triple of is associated with a point of a three-dimensional space whose position is given by , and coordinates.
Each pair of axes in determines a plane. The resulting three planes are called coordinate planes. Each coordinate plane is named after the axes that determine it. Thus, we have the -plane, -plane, and -plane. Coordinate planes intersect at the point , called the origin, and subdivide into eight regions, called octants.
The set of all ordered -tuples , where is a real number for , is called . The point in is called the origin.
cannot be visualized for , but many familiar ideas, such as the distance formula, can be generalized to .
Distance in
In this section we will establish a formula for the distance between two points in . We begin by observing that the distance between two numbers (points) and on the number line is given by . (Why do we use the absolute value brackets?).
We can use the Pythagorean Theorem to establish the distance formula for points of .
Let and be points in . By the Pythagorean Theorem we have Why were we able to drop the absolute value brackets?
The distance formula for points in can also be derived using the Pythagorean Theorem. Let and be points of . Use the following GeoGebra interactive to walk through the steps of the derivation of the distance formula. RIGHT-CLICK and DRAG to rotate the image.
Let points and be projections of and onto the -plane. By the distance formula in , the distance between and is Let . Observe that is a right triangle with , and . By the Pythagorean theorem we have This gives us the following formula.
Observe the similarity of pattern in the distance formulas for , and . We will take advantage of this pattern to define the distance between two points of .
Practice Problems
Consider the equation
2024-09-11 19:04:31 What can be said about all points that satisfy this equation?
Such points are equidistant from the origin. Such points form a four-dimensional
sphere of radius . Such points are located 5 units from the origin. All of the
above.