Area of Parallelogram Formed By Two Vectors

Take a parallelogram formed by the vectors  
\[\vec{a} , \: \vec{b}\]
.
The area of the parallelogram is the magnitude of the cross or vector product of the vectors  
\[\| \vec{a} \times \vec{b} \| = \| \vec{a} \| \| \vec{b} \| sin \theta\]
  where  
\[\theta\]
  is the angle between  
\[\vec{a}\]
  and  
\[\vec{b}\]
.
The cross product is itself a vector, perpendicular to both  
\[\vec{a}\]
  and  
\[\vec{b}\]
  but the area is equal to the magnitude.

area of parallelogram formed by two vectors

Then of course, thye area of the triangle formed by  
\[\vec{a} , \: \vec{b}\]
  is half the area of the parallelogram  
\[\frac{1}{2} \| \vec{a} \| \| \vec{b} \| sin \theta\]
.