Inner Product and Outer Product
The power of vectors (referring here to vectors in two- or three-dimensional space for the time being) lies in the definition of the inner product and the outer product (more often called the cross product, vector product, etc.). Both are operations between two vectors, where the inner product is defined as symmetric and the outer product is defined as antisymmetric. Both satisfy the distributive law.
Following the convention of textbooks, we use \langle, \rangle to denote the inner product and \land to denote the outer product. For the outer product, \times is more commonly used, but to avoid an excess of symbols, we will consistently 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 us calculate the inner product and outer product of two vectors \boldsymbol{A} and \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}A^{\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? That’s it, 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: let \boldsymbol{e}_{\mu} be a set of standard orthonormal bases, then \langle\boldsymbol{e}_{\mu},\boldsymbol{e}_{\nu}\rangle=\delta_{\mu\nu} \tag{3} where \delta_{\mu\nu}=1 when \mu=\nu, and 0 otherwise. In this way, we can calculate the inner product of any two vectors. With this definition, the inner product becomes a tool for determining orthogonality (the inner product of two vectors is 0) and a tool for calculating magnitude (the inner product of a vector with itself).
Now let’s look at the outer product. In two-dimensional space, the outer product is defined as follows: let \boldsymbol{e}_{\mu} be a set of standard orthonormal bases, then \boldsymbol{e}_1\land\boldsymbol{e}_2=1 \tag{4} Note that from antisymmetry, we can derive \boldsymbol{e}_1\land\boldsymbol{e}_1=\boldsymbol{e}_2\land\boldsymbol{e}_2=0 and \boldsymbol{e}_2\land\boldsymbol{e}_1=-1. Thus, this definition is already 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 outer 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, it is defined as: \boldsymbol{e}_1\land\boldsymbol{e}_2=\boldsymbol{e}_3, \quad \boldsymbol{e}_2\land\boldsymbol{e}_3=\boldsymbol{e}_1, \quad \boldsymbol{e}_3\land\boldsymbol{e}_1=\boldsymbol{e}_2 \tag{6} With this definition, the outer product in three-dimensional space is a vector that is perpendicular to the original two vectors, and its magnitude is equal to the area of the parallelogram spanned by the original two vectors.
Reviewing the entire process, we can understand it this way: the inner product and outer product are originally purely algebraic definitions of symmetric and antisymmetric operations. As for their geometric meanings, they depend on the definitions assigned to the inner and outer products of the basis vectors. Of course, the definitions of inner and outer products have certain historical origins, but since the concepts themselves are not particularly difficult, we will omit the study of their history. We can clearly see this line of thought: Pure algebraic definition (mainly defining the inner and outer products of the basis) — Seeking geometric meaning — Reflecting on historical origins.
The Power of Antisymmetry
Since we began learning arithmetic, the operations we have encountered are basically symmetric, i.e., operations satisfying ab=ba. The addition and multiplication of numbers are like this. In high school, when we learned the inner product of vectors, it was still the case. Because not many high schools teach the outer product of vectors, many students do not encounter non-commutative operations (i.e., ab \neq ba) until university, such as matrix multiplication. Among all non-commutative operations, antisymmetric operations are quite special and rich in content. Not only that, they also bring convenience to calculations. Consider the following example.
Consider the problem of a particle moving in a fixed gravitational center; we have the equation of motion: \ddot{\boldsymbol{x}}=-\frac{\mu\boldsymbol{x}}{|\boldsymbol{x}|^3} \tag{7} Taking the outer product with \boldsymbol{x} on both sides: \boldsymbol{x}\land \ddot{\boldsymbol{x}}=-\boldsymbol{x}\land \frac{\mu\boldsymbol{x}}{|\boldsymbol{x}|^3}=0 \tag{8} Noting 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} Then \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, it would represent three equations if written in component form. Thus, through a few simple steps, we have obtained three constants of integration. Reflecting on the root cause, it is the antisymmetry of the outer 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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