## 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.

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