\( % Arcus cosine. \def\acos{\cos^{-1}} % Vector projection. \def\projection#1#2{{proj_{#1}\left(#2\right)}} % Vector rejection. \def\rejection#1#2{{rej_{#1}\left(#2\right)}} % Norm. \def\norm#1{{\left\|#1\right\|}} % Cross product. \def\cross#1#2{\mathit{cross}\left(#1,#2\right)} % Dot product. \def\dot#1#2{{#1 \cdot #2}} % Magnitude. \def\mag#1{{\left|#1\right}} \def\group#1{\left(#1\right)}} \def\sbgrp#1{\left\{#1\right\}} \)

Translate transformation of a line

A line \( \left\{ \mathbf{p} + t \group{\mathbf{p} - \mathbf{q}} | t \in \mathbb{R} \right\} \) subjected to a translate transformation represented by a vector \(\vec{t} \in \mathbb{R}^3\) yields a line \( \left\{ \mathbf{p} + \vec{t} + t \group{\mathbf{p} - \mathbf{q}} | t \in \mathbb{R} \right\} \).