TITLE: Exterior product and four vectors?
QUESTION [0 upvotes]: Quite a straight forward question but something I don't really understand. I have a mediocre grasp on the concept of exterior products in $3$ dimensions. However, is the product defined for two $4$-vectors?  If so how is it defined explicitly given two four vectors?

REPLY [2 votes]: Here is a coordinate description of the exterior product of two vectors in $\mathbb{R}^4$: if $e_i$ denotes the $i$th basis vector, then any $v \in \mathbb{R}^4$ has coordinates $(a^1,a^2 ,a^3,a^4)$, meaning that $v = \sum a^ie_i$. The defining characteristic of the exterior product is that it is bilinear and antisymmetric, so if $w = \sum b^i e_i$ is another vector we must have
$$v \wedge w = \sum_i a^ie_i \wedge \sum_j b^je_j = \sum_{i,j}a^ib^je_i\wedge e_j,$$
and since $e_i \wedge e_j = -e_j \wedge e_i$ this amounts to
$$v\wedge w = (a^1b^2 - a^2b^1)(e_1 \wedge e_2) + \cdots + (a^3b^4 - a^4b^3)(e^3 \wedge e^4)$$
as in Ted Shifrin's answer.
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