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Volume of a Parallelepiped

volume of parallelepiped

For a parallelepiped formed by three vectors  
\[\vec{a}, \: \vec{b}, \: \vec{c}\]
  the area of the base - formed by vectors  
\[\vec{a}, \: \vec{b}\]
  is the magnitude of the vector or cross product of  
\[\vec{a}\]
  and  
\[\vec{b}\]
.
\[Area \: of \: Base = \| \vec{a} \times \vec{b} \|\]

\[\vec{a} \times \vec{b}\]
  is a vector perpendicular to both  
\[\vec{a}\]
  and  
\[\vec{b}\]
).
If the third vector defining the parallelepiped is  
\[\vec{c}\]
  then the volume of the parallelepiped is given by the the area of the base times the projection of  
\[\vec{c}\]
  onto the vector  
\[\vec{a} \times \vec{b}\]
 
\[Volume=\| \vec{a} \times \vec{b} \| \vec{c} \cdot ( \frac{\vec{a} \times \vec{b}}{\| \vec{a} \times \vec{b} \|} )=(\vec{a} \times \vec{b}) \cdot \vec{c}=\left| \begin{array}{ccc} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{array} \right|\]

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