Proof Guide

The majority of proofs of equations relies on either algebraic proofs, geometric proofs, or both to prove a theorem or equation.

# Algebraic Proofs

A large majority of algebraic proofs result from the use of these properties:

## Commutative Property

The commutative property states the arrangement of some multiplied or divided element does not affect the element output. In simple arithmetic:

(1)
\begin{align} ab = ba, \quad a , b \in \mathbb{R} \end{align}

The commutative property does not always hold though. For example, the cross product of two vectors does not follow the commutative property:

(2)
\begin{align} \vec{u} x \vec{v} = - \vec{v} x \vec{u}, \quad \vec{u}, \vec{v} \in \mathbb{R}^3 \end{align}

Therefore:

(3)
\begin{align} \vec{u} x \vec{v} \neq \vec{v} x \vec{u}, \quad \vec{u}, \vec{v} \in \mathbb{R}^3 \end{align}

## Associative Property

The associative property states the arrangement of some added or subtracted element does not affect the element output.

For example with simple arithmetic:

(4)
\begin{align} a + b + c = a + (b + c), \quad a , b, c \in \mathbb{R} \end{align}

Or

(5)
\begin{align} a + b + c = a + c + b, \quad a , b, c \in \mathbb{R} \end{align}

## Distributive Property

The distributive property merges the commutative and associative property. For example with simple arithmetic:

(6)
\begin{align} a(b + c) = ab + ac, \quad a , b, c \in \mathbb{R} \end{align}