Vtyjkyuk

I was shocked to find that the area of parallelogram in two dimensions can be found by cross multiplying two adjacent vectors.

I understand that in three dimensions the area of parallelogram is the cross product of two adjacent vectors $$|a times b|$$ but no text books have explained why in two dimensions cross product of two adjacent vectors are the area of a parallogram. Could someone explain?

Basically, because it is true in dimension $3$. So, if $v=(a,b)$ and $w=(c,d)$ are two vectors of $mathbbR^2$, consider $v^star=(a,b,0)$ and $w^star=(c,d,0)$. Then $v^star,w^starinmathbbR^3$ and the area of the parallelogram spanned by $v$ and $w$ is equal to the area of the parallelogram spanned by $v^star$ and $w^star$, which is $|v^startimes w^star|$.

Another way of looking at it: The dot product of two vectors, u and v, is given by $|u||v|sin(theta)$ where $theta$ is the angle between the two vectors. The area of a parallelogram is "base times height" where the "height" is measured perpendicular to "base". If we take v to be the base, dropping a perpendicular from the tip of u to v, we have a right triangle with angle $theta$ and hypotenuse |u|. The height, the "opposite side" is given by $|u|sin(theta)$ so the area of the parallelogram, "base times height", is $|v|(|u|sin(theta))= |u||v|sin(theta)$, the dot product.