Sets of Ordered Triples

Section 1.3 Sets of Ordered Triples

Checkpoint 1.3.1.

Give geometric representations for the set defined by

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

both using point and using vector interpretations. (Unless otherwise indicated start vectors at the origin.)

Definition 1.3.2.

A line extends coninuously and infinitely in one direction and the exact opposite of that direction. This means points lie on that line at every distance in that direction or its opposite from any point on that line.

Definition 1.3.3.

A plane extends continuously and infinitely in any combination of two directions, including the exact opposite of those two directions. This means points lie on that plane at every combination of those directions from any point on the plane.

Checkpoint 1.3.4.

Give geometric representations for the set defined by

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

both using point and using vector interpretations.

Checkpoint 1.3.5.

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 1.3.6.

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 1.3.7.

For any three points \(\vect{v}\text{,}\) \(\vect{w}\) and \(\vect{u}\text{,}\) such that \(\vect{v} \neq \vect{w}\text{,}\) and \(\vect{u}\) is not on the line through \(\vect{v}\) and \(\vect{w}\text{,}\) there exists a set of points describing a plane which passes through points \(\vect{v}\text{,}\) \(\vect{w}\) and \(\vect{u}\text{.}\)