A Brief Introduction to \(\RR ^n\)

The set of all real numbers is denoted by \(\RR \). It is convenient to associate real numbers with points on a line, called the real number line.

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The set of all ordered pairs \((x, y)\), where \(x\) and \(y\) are real numbers, is called \(\RR ^2\). Using set notation we write:

\[\RR ^2=\{(x, y):x,y\in \RR \}\]

Geometrically speaking, \(\RR ^2\) can be associated with a coordinate plane in which we refer to each point by its \(x\) and \(y\) coordinates.

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The set of all ordered triples \((x, y, z)\), where \(x\), \(y\) and \(z\) are real numbers, is called \(\RR ^3\).

\[\RR ^3=\{(x, y, z):x,y, z\in \RR \}\]

Geometrically, an ordered triple of \(\RR ^3\) is associated with a point of a three-dimensional space whose position is given by \(x\), \(y\) and \(z\) coordinates.

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Each pair of axes in \(\RR ^3\) 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 \(xy\)-plane, \(xz\)-plane, and \(yz\)-plane. Coordinate planes intersect at the point \((0, 0, 0)\), called the origin, and subdivide \(\RR ^3\) into eight regions, called octants.

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The set of all ordered \(n\)-tuples \((x_1, x_2, \ldots , x_n)\), where \(x_i\) is a real number for \(1\leq i\leq n\), is called \(\RR ^n\).

\[\RR ^n=\{(x_1, x_2,\ldots ,x_n)|x_i\in \RR \, \text {for}\, 1\leq i\leq n\}\]

The point \((0,0,\ldots , 0)\) in \(\RR ^n\) is called the origin.

\(\RR ^n\) cannot be visualized for \(n>3\), but many familiar ideas, such as the distance formula, can be generalized to \(\RR ^n\).

Distance in \(\RR ^n\)

In this section we will establish a formula for the distance between two points in \(\RR ^n\). We begin by observing that the distance between two numbers (points) \(x_1\) and \(x_2\) on the number line is given by \(|x_1-x_2|\). (Why do we use the absolute value brackets?).

We can use the Pythagorean Theorem to establish the distance formula for points of \(\RR ^2\).

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Let \(A(x_1, y_1)\) and \(B(x_2, y_2)\) be points in \(\RR ^2\). By the Pythagorean Theorem we have

\[AB^2=(x_1-x_2)^2+(y_1-y_2)^2\]
\[AB=\sqrt {(x_1-x_2)^2+(y_1-y_2)^2}\]

Why were we able to drop the absolute value brackets?

The distance formula for points in \(\RR ^3\) can also be derived using the Pythagorean Theorem. Let \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\) be points of \(\RR ^3\). 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 \(A'(x_1, y_1, 0)\) and \(B'(x_2, y_2,0)\) be projections of \(A\) and \(B\) onto the \(xy\)-plane. By the distance formula in \(\RR ^2\), the distance between \(A'\) and \(B'\) is

\[A'B'=\sqrt {(x_1-x_2)^2+(y_1-y_2)^2}\]

Let \(C=(x_2, y_2, z_1)\). Observe that \(\triangle {ABC}\) is a right triangle with \(AC=A'B'\), and \(BC=|z_1-z_2|\). By the Pythagorean theorem we have

\[AB^2=AC^2+BC^2=\left (\sqrt {(x_1-x_2)^2+(y_1-y_2)^2}\right )^2+(z_1-z_2)^2=(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2\]

This gives us the following formula.

Observe the similarity of pattern in the distance formulas for \(\RR ^1\), \(\RR ^2\) and \(\RR ^3\). We will take advantage of this pattern to define the distance between two points of \(\RR ^n\).

Practice Problems

Find the coordinates of each point.

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Answer:

\[P(\answer {5}, \answer {5}, \answer {10})\]
\[Q(\answer {-6}, \answer {3}, \answer {2})\]
\[R(\answer {3}, \answer {-5}, \answer {-8})\]
Find the coordinates of each point. RIGHT-CLICK and DRAG to rotate the image.

Answer:

\[A(\answer {-1}, \answer {4}, \answer {2})\]
\[B(\answer {3}, \answer {2}, \answer {-3})\]
\[C(\answer {2}, \answer {-5}, \answer {3})\]
\[D(\answer {-3}, \answer {-2}, \answer {4})\]
Find the distance between \(A(-2, -1, 4)\) and \(B(1, -5, -8)\).
\[AB=\answer {13}\]