what is the difference between cross product and exterior product?

Question

I have learn that the exterior product is an oriented plane called bivector given as $A \times B = |A||B| \sin x (i \times j)$ For $x \in(-\pi,\pi)$. I will like someone to derive the cross product and the exterior product with a geometry picture to show the difference.

Answer 1

I believe the exterior products (wedge products) of two vectors $u$ and $v$ are denoted by $u\wedge v$. Note: the $LaTeX$ command for wedge products is also \wedge.

I have borrowed this picture from Wikipedia's page on exterior algebra as it explains the difference very clearly. $a\times b$ is the cross product of the two vectors $a$ and $b$. The cross product gives a vector that is perpendicular to both $\vec{a}$ and $\vec{b}$ as you can see in the diagram. This is particularly useful in physics where we encounter several cross products in nature: torque $\tau=\vec{r}\times \vec{F}$ where $\vec{r}$ is the radius from the axis of rotation and $\vec{F}$ is the force, force on a moving charge $\vec{F}=q\vec{v}\times\vec{B}$ where $\vec{B}$ is the magnetic field, and so on.

Thus, this led to the "right-hand rule as we want to find the direction of the vector cross product. Note: in the picture, even a vector anti-parallel to the vector $a\times b$ would be perpendicular to both vectors $a$ and $b$.

Imagine: $\vec{a}$ as your thumb and point your thumb in the direction of $\vec{a}$. $\vec{b}$ as your fingers and point them in the direction of $\vec{b}$. $\vec{a}\times \vec{b}$ is in the direction that your palm is pointing.

On the other hand, the wedge product is used to determine areas, volumes, etc, as shown by $a\wedge b$, which shows the area of a parallelogram. This is exactly the formula that you have provided us as it determines the area of the parallelogram from two vectors $\vec{A}$, $\vec{B}$, and the angle between $\vec{A}$ and $\vec{B}$.

Note: the magnitude of the cross product as shown below also gives the area of a parallelogram. Of course, this is different from taking only the cross product itself as we have the magnitude (length of a vector) here instead. $$||a\times b||=||a||\quad\!\!\!||b||\quad\!\!\!|\sin\theta|$$

Answer 2

I don't know what you mean by derive here, but by definition the two products give different types of geometric objects. The exterior product of two vectors (in $\mathbf{R}^n$, for any $n$) is a bivector, whereas the cross product of two vectors (in $\mathbf{R}^3$ only) is another vector. For $n=3$, there happens to be an isomorphism between bivectors and vectors, and the cross product is what you get if you take the exterior product and the convert then resulting bivector to a vector using this isomorphism.