Sets of Ordered Triples

Section 1.3 Sets of Ordered Triples

Checkpoint 1.3.1.

Describe the following set geometrically.

\begin{equation*} \mathbb{S}=\{\ t\ot{-1}{2}{0} \ |\ t~\in~\mathbb{R}\ \} \end{equation*}

Definition 1.3.2.

A line is a nonempty set \(\mathbb{L} \subseteq \mathbb{R}^3\) where a single direction \(\vect{u}\text{,}\) or its opposite \(-\vect{u}\text{,}\) describes the movement between any two points in \(\mathbb{L}\text{.}\) In addition \(\mathbb{L}\) is not a proper subset of any other set with this property.

Checkpoint 1.3.3.

Describe the following set geometrically.

\begin{equation*} \mathbb{T} = \{\ \ot{3}{1}{0} + t\ot{-1}{2}{0}\ |\ t \in \mathbb{R}\ \} \end{equation*}

Checkpoint 1.3.4.

Describe the set of points on the line in \(\mathbb{R}^3\) that passes through points \(\vect{v} = \ot{1}{2}{3}\) and \(\vect{w} = \ot{2}{0}{-2}\text{.}\)

Theorem 1.3.5.

For any two points \(\vect{v}\) and \(\vect{w}\) such that \(\vect{v}\neq \vect{w}\) there exists a set of points describing a line passing through both \(\vect{v}\) and \(\vect{w}\text{.}\)

Challenge 1.3.6.

For any two points \(\vect{v}\) and \(\vect{w}\) such that \(\vect{v}\neq \vect{w}\) the set of points describing a line passing through both \(\vect{v}\) and \(\vect{w}\) is unique.