# Relation between cross-product and outer product

If 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, but yet the outer product of two column vectors in $\mathbb{R}^3$ is a 3x3 matrix, not another column vector. What's the link? Thanks!

• – joriki Aug 13 '12 at 10:17
• Just found this, on the ambiguity of the term "outer product": en.wikipedia.org/wiki/… – Ryan G Aug 13 '12 at 11:28

## 2 Answers

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? – Ryan G Aug 13 '12 at 10:45
• I think it is naming conventions ambiguity. In your reference outer product is exactly what I call exterioir product. – Norbert Aug 13 '12 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?? – Ryan G Aug 13 '12 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. – Norbert Aug 13 '12 at 11:15
• Thanks Norbert. Is this map fair? math.stackexchange.com/q/182024/21813 – Ryan G Aug 13 '12 at 12:13

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 Jun 9 '16 at 13:42