Area of parallogram in two dimensions using cross product

Question

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?

Answer 1

Basically, because it is true in dimension $3$. So, if $v=(a,b)$ and $w=(c,d)$ are two vectors of $\mathbb{R}^2$, consider $v^\star=(a,b,0)$ and $w^\star=(c,d,0)$. Then $v^\star,w^\star\in\mathbb{R}^3$ 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 $\|v^\star\times w^\star\|$.

Answer 2

Another way of looking at it: The cross 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 cross product.