Relationship between cross product and outer product

Since inner products $$(V)$$ are generalisations of dot products $$(\mathbb{R}^n),$$ then are outer products $$(V)$$ also related to cross products $$(\mathbb{R}^3)$$ in some way?

A quick search reveals that they are—yet the outer product of two column vectors in $$\mathbb{R}^3$$ is a $$3\times3$$ matrix, not another column vector. What is the connection?

• Commented Aug 13, 2012 at 10:17
• Just found this, on the ambiguity of the term "outer product": en.wikipedia.org/wiki/… Commented Aug 13, 2012 at 11:28
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Commented Jun 22, 2022 at 14:17
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Cross product is much more related to exterior product which is in fact a far going generalization.

Outer product is a matricial description of tensor product of two vectors.

• Ah, thanks! So are outer products and exterior products related in some way then? In euclideanspace.com/maths/algebra/clifford/theory/extendXProduct/… , the author claims that "Inner product by a vector reduces the grade of a multivector. It is related to the dot product. Outer product by a vector increases the grade of a multivector. It is related to the cross product." and I've seen this claim elsewhere too. Are they mixing up their terminologies? Commented Aug 13, 2012 at 10:45
• I think it is naming conventions ambiguity. In your reference outer product is exactly what I call exterioir product. Commented Aug 13, 2012 at 10:49
• en.wikipedia.org/wiki/Interior_product states that "The interior product, named in opposition to the exterior product ... should not be confused with an inner product." So Inner is a generalisation of Dot. Exterior is a generalisation of Cross. Interior is in opposition to Exterior but shld not be confused with Inner. Can someone put these into a simple framework for me? For eg. how is Inner related to Interior then?? Commented Aug 13, 2012 at 10:57
• Interior product is degree descending operation. It looks like inner product, but this is just an analogy. Here is an example of how one can define interior product given some $x\in V$. You define interioir product of vectors $x_1,\ldots,x_{n-1}$ as exterior product of $x, x_1,\ldots, x_n$. Note that interioir product depends on the choice of vector $x$. Interioir products are closely related to annihilation operators on the Fock space in quantum mechanics. Commented Aug 13, 2012 at 11:15
• Thanks Norbert. Is this map fair? math.stackexchange.com/q/182024/21813 Commented Aug 13, 2012 at 12:13

In Geometric algebra, the cross-product of two vectors is the dual (i.e. a vector in the orthogonal subspace) of the outer product of those vectors in $$\mathbb{G}^3$$ (so in a way you could say that the outer product generalizes the dot product, although the cross product is not an outer product). This is very well explained in the great introduction series to Geometric Algebra by Alan Macdonald. The relation between the cross product and the outer product is mentioned in Geometric Algebra 4 at min 6:20.

Now Geometric algebra is just Clifford algebra, with a different emphasis, so that these definitions are equivalent to the definitions in that algebra.

The inner product creates a scalar and the outer a skew-symmetric matrix. If you sum like-terms of this matrix you get a vector which also results from the cross product computation.

• What do you mean by sum like-terms of this matrix? How to sum?
– Royi
Commented Jun 9, 2016 at 13:42
• Outer products don't produce skew-symmetric matrices, except in very niche cases like a zero matrix as the output matrix of two zero vectors (which is technically skew-symmetric). Commented Jul 19 at 22:21