Properties of the Vector or Cross Product

The vector or cross product of two vectors has several useful properties.
The vector product of two vectors is itself a vector, and is perpendicular to the two vectors.
If the ngle between vectors  
  so that vector  
  is 90 degrees anticlockwise from  
, then the vectors  
\[\vec{a}, \: \vec{b}, \: \vec{a} \times \vec{b}=\vec{c}\]
  forms a right handed coordinates system.
\[\vec{a} \times k \vec{a}=0\]

\[\vec{a} \times \vec{b}=\| \vec{a} \| \| \vec{b}, \| \vec{a}. \| vec{b} \neq \vec{0} \|\]
\[\vec{a}, \: \vec{b}\]
  are perpendicular, and vice versa.
\[\vec{a} \times \vec{b}= - \vec{b} \times \vec{a}\]

\[\| \vec{a} \times \vec{b} \|= \| \vec{b} \times \vec{a} \|\]

\[\vec{a} \cdot (\vec{b} \times \vec{b})=\left| \begin{array}{ccc} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{array} \right| \]
  is the vector triple product. It is a number, with magnitude equal to the volume of the parallelpiped formed by the vectors
\[\vec{a}, \: \vec{b}, \: \vec{c}\]
\[\vec{a} \times (\vec{b}+\vec{c})=(\vec{a} \times \vec{b})+(\vec{a} \times \vec{c})\]

\[(\vec{a}+\vec{b}) \times (\vec{c}+\vec{d})=(\vec{a} \times \vec{c})+(\vec{a} \times \vec{d})+(\vec{b} \times \vec{c})+(\vec{b} \times \vec{d})\]

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