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Section 1.1 Ordered Triples

Definition 1.1.1.

An ordered triple of real numbers is written \(\vect{v}=\ot{v_1}{v_2}{v_3}\) where \(v_1\text{,}\) \(v_2\text{,}\) \(v_3\) \(\in \mathbb{R}\text{.}\) Such a triple can be interpreted geometrically as either a point: indicating a position in 3 dimensional space, or as a vector: visualized as an arrow indicating direction which can occur at any position.

Unless otherwise indicated, the Cartesian coordinate system, also known as the rectangular coordinate system, will be used. In this system the x,y and z axis are perpendicular and indicate the direction for the first, second and third coordinates respectively.

The xyz-axes.
Figure 1.1.2.

As a point in the Cartesian coordinate system, \(\vect{v} = \ot{v_1}{v_2}{v_3}\text{,}\) is the position reached by beginning where the axes meet and moving a distance of \(v_{1}\) in the x direction, then \(v_2\) in the y direction followed by \(v_3\) in the z direction.

As a vector in the Cartesian coordinate system, \(\vect{v} = \ot{v_1}{v_2}{v_3}\text{,}\) is visualized as an arrow which can start at any position and ends at the point obtained by moving a distance of \(v_1\) in the x direction, then \(v_2\) in the y direction followed by \(v_{3}\) in the z direction.

Ordered triples can represent physical concepts such as position, velocity or force. They can also represent any triple of related data such as population of three related species: mice, snakes and hawks, or three related physical properties: temperature, pressure and volume.

Definition 1.1.3.

The set of all ordered triples is called Euclidean 3-space and written: \(\mathbb{R}^3\text{.}\) Using set notation it is written:

\begin{equation*} \mathbb{R}^3=\{\ot{v_1}{v_2}{v_3} | v_1,v_2,v_3 \in \mathbb{R}\} \end{equation*}
Definition 1.1.4.

Let \(\vect{v}=\ot{v_1}{v_2}{v_3} \in \mathbb{R}^3\text{.}\) The norm, also called length or the magnitude and denoted \(\|\vect{v}\|\text{,}\) is the distance of the point from the origin when \(\vect{v}\) is interpreted as a position and the length of the arrow when \(\vect{v}\) is interpreted as a vector.

Give geometric representations of the points \(\vect{v} = \ot{3}{-4}{0}\) and \(\vect{w} = \ot{3}{-4}{2}\) using both point and vector interpretations.

For \(\vect{v} = \ot{3}{-4}{0}\) and \(\vect{w} = \ot{3}{-4}{2}\) find \(\|\vect{v}\|\) and \(\|\vect{w}\|\text{.}\) Show and justify each step.

Consider an ordered triple \(\vect{v}=\ot{v_1}{v_2}{v_3}\text{,}\) with \(\|\vect{v}\|=10\text{.}\) Does there exist such a \(\vect{v}\text{?}\) If so, is \(\vect{v}\) unique? In other words is there exactly one such \(\vect{v}\text{?}\)