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

## Uniform scale transformation of a ray

A ray $$(O,\vec{d})$$ in $$\mathbb{R}^3$$ subjected to a uniform scale transformation represented by a scalar $$s \in \mathbb{R}$$ yields a ray $$(O',\vec{d}')$$ where $$O' := s O$$ and $$\vec{d}' := s (\vec{d} + O) - O'$$.

## Scale transformation of a ray

An ray $$(O,\vec{d})$$ in $$\mathbb{R}^3$$ subjected to a scale transformation represented by a vector $$\vec{s} \in \mathbb{R}^3$$ yields a ray $$(O', \vec{d})$$ where $$O' := \vec{s} \cdot O$$ and $$\vec{d}' := \vec{d} \cdot (\vec{d} + O) - O'$$.

## Translate transformation of a ray

A ray $$(O,\vec{d})$$ in $$\mathbb{R}^3$$ subjected to translate transformation represented by a vector $$\vec{t} \in \mathbb{R}^3$$ yields a ray $$(O',\vec{d}')$$ where $$O' := O + \vec{t}$$ and $$\vec{d}' := \vec{d}$$.