The vectors $\langle 2, 0,0\rangle$ and $\langle 2,1,0 \rangle$ form a triangle with the area $\dfrac{base \times height}{2}$. However normal vectors are supposed to tell what the area is too, but the normal vector that I just found through wolframalpha is $\langle 0,0,2\rangle$.
$\dfrac{base \times height}{2}$ equals $\dfrac{2 \times 1}{2} = 1$.
What step am I doing wrong here!?
On
Area is considered to be a vector quantity in physics whose direction is perpendicular to both sides of the triangle. The magnitude of area is calculated as the magnitude of the vector product or cross product of both vectors. If we consider the cross product we will get the magnitude as |A||B|sin(θ). But if we take in consideration the fact that |A| and |B|sin(θ) are in fact the magnitude of the base and height of the parallelogram formed by vector A and B then their product will be the area of the parallelogram. Since we know that the area of a triangle is half the area of a parallelogram thus we can simply use half of the cross product of the two vectors to find the area of the triangle formed by them.The following image explains the direction of the area
The magnitude of the cross product is the area of the parallelogram with the vectors for sides.
You should multiply by $\frac12$ to get the area of the triangle.