Why is this the area? Suppose that we have a triangle with vertices the points $0, \vec{a}, \vec{b}$.
Why does it hold that the area of the triangle is equal to the half of the area of the parallelogram with sides $\vec{a}$ and $\vec{b}$ ?

Look at this image:

Suppose the sides labeled $a$ and $b$ are the two vectors.  You can see the triangle is exactly half the area of the parallelogram.  If you cut the part to the left of the dashed line and glue it on to the right side you'll have a perfect rectangle.  The area of that rectangle is then obviously $b\times h$ the same as the area of the parallelogram.    So the area of the triangle is exactly half of that.  I hope this helps!
