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

## Translate transformation of a line segment

A line segment $$\left\{ \mathbf{p} + t \group{\mathbf{p} - \mathbf{q}} | t \in \mathbb{R}, 0 \leq t \leq 1 \right\}$$ in $$\mathbb{R}^3$$ subjected to a translate transformation represented by a vector $$\vec{t} \in \mathbb{R}^3$$ yields a line segment $$\left\{ \mathbf{p} + \vec{t} + t \group{\mathbf{p} - \mathbf{q}} | t \in \mathbb{R}, 0 \leq t \leq 1 \right\}$$.