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

## 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 \( \left\{ \mathbf{p} + \vec{t} + t \group{\mathbf{p} - \mathbf{q}} | t \in \mathbb{R} \right\} \).