By 苏剑林 | November 04, 2016
The power of vectors (referring here temporarily to vectors in two-dimensional or three-dimensional space) lies in the definition of inner products and exterior products (more often called cross products, vector products, etc.). Both are operations between two vectors, where the inner product is defined to be symmetric, while the exterior product is defined to be antisymmetric, and both satisfy the distributive law.
Following the tradition of textbooks, we use $\langle,\rangle$ to denote the inner product and $\land$ to denote the exterior product. For the exterior product, $\times$ is more commonly used, but to avoid an excess of symbols, we will uniformly use $\land$. We write vectors in terms of their basis, for example $$\boldsymbol{A}=\boldsymbol{e}_{\mu}A^{\mu} \tag{1} $$ where $\boldsymbol{e}_{\mu}$ represents a set of basis vectors, and $A^{\mu}$ represents the components of the vector. Let's calculate the inner and exterior products of two vectors $\boldsymbol{A}, \boldsymbol{B}$: $$\begin{aligned}&\langle \boldsymbol{A}, \boldsymbol{B}\rangle=\langle \boldsymbol{e}_{\mu}A^{\mu}, \boldsymbol{e}_{\nu}B^{\nu}\rangle=\langle\boldsymbol{e}_{\mu},\boldsymbol{e}_{\nu}\rangle A^{\mu}B^{\nu}\\ &\boldsymbol{A}\land \boldsymbol{B}=(\boldsymbol{e}_{\mu}A^{\mu})\land (\boldsymbol{e}_{\nu}B^{\nu})=\boldsymbol{e}_{\mu}\land\boldsymbol{e}_{\nu} A^{\mu}B^{\nu} \end{aligned} \tag{2} $$
And then? There is no "and then," because we haven't yet defined $\langle\boldsymbol{e}_{\mu},\boldsymbol{e}_{\nu}\rangle$ and $\boldsymbol{e}_{\mu}\land\boldsymbol{e}_{\nu}$. In analytic geometry, we define the inner product as follows: if $\boldsymbol{e}_{\mu}$ is a set of standard orthonormal bases, then $$\langle\boldsymbol{e}_{\mu},\boldsymbol{e}_{\nu}\rangle=\delta_{\mu\nu} \tag{3} $$ When $\mu=\nu$, $\delta_{\mu\nu}=1$, otherwise it is 0. In this way, we can calculate the inner product for any two vectors. With this definition, the inner product becomes a tool for judging orthogonality (the inner product of two vectors is 0) and a tool for calculating norm (a vector's inner product with itself).
Looking at the exterior product, in two-dimensional space, it is defined as follows: if $\boldsymbol{e}_{\mu}$ is a standard orthonormal basis, then $$\boldsymbol{e}_1\land\boldsymbol{e}_2=1 \tag{4} $$ Note that from antisymmetry, we can obtain $\boldsymbol{e}_1\land\boldsymbol{e}_1=\boldsymbol{e}_2\land\boldsymbol{e}_2=0$ and $\boldsymbol{e}_2\land\boldsymbol{e}_1=-1$, so this definition is complete. At this point, we can calculate: $$\boldsymbol{A}\land \boldsymbol{B}=A^1 B^2 - A^2 B^1 \tag{5} $$ In this case, the exterior product is a scalar, and its absolute value is precisely the area of the parallelogram spanned by $\boldsymbol{A}$ and $\boldsymbol{B}$.
In three-dimensional space, the definition is: $$\boldsymbol{e}_1\land\boldsymbol{e}_2=\boldsymbol{e}_3, \boldsymbol{e}_2\land\boldsymbol{e}_3=\boldsymbol{e}_1, \boldsymbol{e}_3\land\boldsymbol{e}_1=\boldsymbol{e}_2 \tag{6} $$ Under this definition, the exterior product in three-dimensional space results in a vector which is perpendicular to the original two vectors, and its magnitude equals the area of the parallelogram spanned by the two original vectors.
Reviewing the whole process, we can understand it this way: inner and exterior products are essentially pure algebraic definitions of symmetric and antisymmetric operations. As for their geometric meanings, they depend on the meanings further assigned after the inner and exterior products of the basis vectors are determined. Of course, the definitions of inner and outer products have certain historical origins, but since the concept itself is not difficult, we will omit the research into its history. It can be seen that for the inner product, the definition can clearly be generalized to higher-dimensional space, whereas the exterior product is not as straightforward. Regardless, we can clarify this line of thought: Pure algebraic definition (mainly defining the products of basis vectors) — Seeking geometric meaning — Reviewing historical origins.
From the time we start learning arithmetic, the operations we encounter are basically symmetric, i.e., operations that satisfy $ab=ba$. The addition and multiplication of numbers are like this. In high school, when studying the inner product of vectors, it is still like this. Because not many high schools actually teach the exterior product of vectors, many students do not encounter non-commutative operations (where $ab \neq ba$) until university, such as matrix multiplication. Among all non-commutative operations, the antisymmetric operation is a special type with quite rich content. Not only that, it also brings convenience to calculations. Consider the following example.
Consider the problem of a particle moving in a fixed gravitational center; then we have the equation of motion: $$\ddot{\boldsymbol{x}}=-\frac{\mu\boldsymbol{x}}{|\boldsymbol{x}|^3} \tag{7} $$ Taking the exterior product of both sides with $\boldsymbol{x}$, we get: $$\boldsymbol{x}\land \ddot{\boldsymbol{x}}=-\boldsymbol{x}\land \frac{\mu\boldsymbol{x}}{|\boldsymbol{x}|^3}=0 \tag{8} $$ Notice that $$\frac{d}{dt}(\boldsymbol{x}\land \dot{\boldsymbol{x}}) = \dot{\boldsymbol{x}}\land \dot{\boldsymbol{x}}+\boldsymbol{x}\land \ddot{\boldsymbol{x}}=\boldsymbol{x}\land \ddot{\boldsymbol{x}} \tag{9} $$ So the above equation implies $$\frac{d}{dt}(\boldsymbol{x}\land \dot{\boldsymbol{x}})=0 \tag{10} $$ Therefore $$\boldsymbol{x}\land \dot{\boldsymbol{x}}=\boldsymbol{C} \tag{11} $$ This is in fact the conservation of angular momentum. Since it is a vector equation, if written in component form, it would yield three equations. Thus, through a few simple steps, we have obtained three constants of integration. Reflecting on the root cause, it is the antisymmetry of the exterior product $a\land b = -b\land a$ that determines $a\land a=0$. This is a property of any antisymmetric quantity and is the power of antisymmetry—it naturally eliminates many parts that ought to be zero.
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,
author={Su Jianlin},
year={2016},
month={Nov},
url={\url{https://kexue.fm/archives/4054}},
}