Vector (or Cross) Product: Formulas, Properties & Area of triangle and Parallelogram

Formula

$$ \boxed{ \textbf{a} \times \textbf{b} = |\textbf{a} | | \textbf{b} | \sin \theta \phantom{0} \hat{ \textbf{n} } } $$

If vector $\textbf{a} = \left( \begin{matrix} a_1 \\ a_2 \\ a_3 \end{matrix} \right) $ and vector $\textbf{b} = \left( \begin{matrix} b_1 \\ b_2 \\ b_3 \end{matrix} \right) $, the vector product of $\textbf{a}$ and $\textbf{b}$ produces a vector perpendicular to both $\textbf{a}$ and $\textbf{b}$ $$ \boxed{ \textbf{a} \times \textbf{b} = \left( \begin{matrix} a_1 \\ a_2 \\ a_3 \end{matrix} \right) \times \left( \begin{matrix} b_1 \\ b_2 \\ b_3 \end{matrix} \right) = \left( \begin{matrix} a_2 b_3 - a_3 b_2 \\ a_3 b_1 - a_1 b_3 \\ a_1 b_2 - a_2 b_1 \end{matrix} \right) \phantom{.} } $$

Parallel vectors

If vectors $\textbf{a}$ and $\textbf{b}$ are parallel, $$ \boxed{ \textbf{a} \times \textbf{b} = 0 } $$

Properties

Non-commutative property: $$ \boxed{\textbf{a} \times \textbf{b} = -\textbf{b} \times \textbf{a}} $$

Distributive property: $$ \boxed{ \textbf{a} \times (\textbf{b} + \textbf{c}) = \textbf{a} \times \textbf{b} + \textbf{a} \times \textbf{c} }$$ $$ \boxed{ (\textbf{b} + \textbf{c}) \times \textbf{a} = \textbf{b} \times \textbf{a} + \textbf{c} \times \textbf{a} } $$

Associative property: If $\lambda$ is a constant, $$ \boxed{ \textbf{a} \times (\lambda \textbf{b}) = (\lambda \textbf{a}) \times \textbf{b} = \lambda (\textbf{a} \times \textbf{b} ) } $$

Value of $ \textbf{a} \times \textbf{a} $: $$ \boxed{ \textbf{a} \times \textbf{a} = 0 } $$

Area of triangle

$$ \boxed{ \text{Area of triangle } OAB = {1 \over 2} \left| \overrightarrow{OA} \times \overrightarrow{OB} \right| \phantom{.} } $$

Area of parallelogram

$$\boxed{ \text{Area of parallogram } OABC = \left| \overrightarrow{OA} \times \overrightarrow{OC} \right| \phantom{.} }$$