Need more work on bivectors here; if there is a lot more to write about on p-vectors, then maybe the bivector stuff should be shifted to a new article.--- Mpatel 08:50, 27 July 2005 (UTC) reply

Yep,

There must be a mistake here :

${\displaystyle F\in \Lambda ^{2}V}$ in terms of a basis ${\displaystyle e_{a}\wedge e_{b}}$ of ${\displaystyle \Lambda ^{2}V}$

${\displaystyle e_{a}\wedge e_{b}}$ is not a basis of ${\displaystyle \Lambda ^{2}V}$ because these 2-vectors are not linearly independant. If we take ${\displaystyle e_{1}\wedge e_{2}}$ and ${\displaystyle e_{2}\wedge e_{1}}$ these 2 2-vectors are linked by the relation ${\displaystyle e_{1}\wedge e_{2}}$ = - ${\displaystyle e_{2}\wedge e_{1}}$.

A good basis for ${\displaystyle \Lambda ^{2}V}$ should be ${\displaystyle e_{a}\wedge e_{b}}$ with ${\displaystyle 1\leq a<b\leq n}$

Shouldn't the name of the lemma be "multivecotor" instead of "p-vector"? -- Trigamma 15:23, 4 November 2007 (UTC) reply

Having a closer look at the article, it should rather be "bivecors", for the article doesn't say anything non-trivial about p-vecors with p \ne 2. —Preceding unsigned comment added by Trigamma ( talk • contribs) 15:32, 4 November 2007 (UTC) reply

Honestly, the entire entry for P-Vector makes no sense to anyone that is not a specialist in tensorial math. And if someone is a specialist in tensor math, why would they be reading this section. This section needs a rewrite into a form that actually conveys a description without a complete reliance on other abstract concepts. —Preceding unsigned comment added by 74.249.92.155 ( talk) 16:37, 24 January 2009 (UTC) reply

Seems that way to me too; also no in-line references. Brews ohare ( talk) 19:23, 21 December 2009 (UTC) reply

Here is a proposal for a new section:

Bivector in three-dimensional Euclidean space

If a and b are any vectors in a real three-dimensional Euclidean vector space, then the complex vector A given by:

${\displaystyle {\boldsymbol {A}}={\boldsymbol {a}}+i{\boldsymbol {b}}\ ,}$

is a bivector, where i = √−1. Two bivectors A = a + i b and A′ = a′ + i b′ are equal if and only if a = a′ and b = b′. ^[1]

Given an orthonormal triad of real unit vectors i, j, k,

${\displaystyle {\boldsymbol {A}}=A_{1}\ {\boldsymbol {i}}+A_{2}\ {\boldsymbol {j}}+A_{3}\ {\boldsymbol {k}}\ }$

${\displaystyle =(a_{1}+i\ b_{1})\ {\boldsymbol {i}}+(a_{2}+i\ b_{2})\ {\boldsymbol {j}}+(a_{3}+i\ b_{3})\ {\boldsymbol {k}}\ .}$

Introducing another bivector B, dot and cross products are: ^[1]

${\displaystyle {\boldsymbol {A\cdot B}}=A_{1}B_{1}+A_{2}B_{2}+A_{3}B_{3}\ ,}$

${\displaystyle {\boldsymbol {A\times B}}=(A_{2}B_{3}-A_{3}B_{2})\ {\boldsymbol {i}}+(A_{3}B_{1}-A_{1}B_{3})\ {\boldsymbol {j}}+(A_{1}B_{2}-A_{2}B_{1})\ {\boldsymbol {k}}\ .}$

Unit bivectors { e_ℓ } are related to the imaginary i=√-1 using the connection (for example): ^[2]

${\displaystyle ({\boldsymbol {e_{1}}}\wedge {\boldsymbol {e_{2}}})^{2}=-1\ ,}$

connecting this discussion to that above, which does not use the imaginary i. See also these historical remarks. ^[3]

Brews ohare ( talk) 19:04, 21 December 2009 (UTC) reply

Sorry, I didn't notice this on the talk page before removing this section or I would have come here first. But those are not bivectors. I had a look at the Boulange+ Hayes source before reverting and what they seem to be describing is something completely different: a way of packaging two vectors as a single object, which you can do stuff with but it's not a bivector, it just happens to have the same name. Perhaps this is clearest as their A+iB objects are six dimensional, bivectors in three dimensions are only three dimensional - they correspond to pseudovectors in 3D. P85 of the second ref above is talking about bivectors in ℝ², It's later (and not in the online preview) they talk about bivectors in 3D.

I've got some thoughts on how to expand it in line with my own understanding, using refs I have here. The bivector stuff should be easy to fix with good examples in 2, 3 and 4 dimensions, and there's probably something interesting to add about trivectors too. -- JohnBlackburne ( talk) 19:29, 21 December 2009 (UTC) reply

I hope no too tendentiously, I changed the redirection from bivector to p-vector and wrote a starting article for bivector at Bivector in accordance with some objections here that the general case was too abstruse for the uninitiated. Perhaps you could take a look at that and make any necessary changes. Brews ohare ( talk) 21:09, 21 December 2009 (UTC) reply

I've added some stuff and clarified some other stuff there. Still a lot more that could be added here and there - if we've two articles we should try and keep duplication to a minimum or one or the other will be nominated for removal/merge. I like that you've used David Hestenes's book as reference, it's one of the two books I have on GA and a remarkable work. -- JohnBlackburne ( talk) 21:54, 21 December 2009 (UTC) reply

The restriction of the term p-vector to refer only to differential geometry seems too narrow. It also has meaning in linear algebra. For example, a bivector also is called sometimes a 2-vector, and is not restricted to differential geometry. Perhaps the use of p-vector as an adjective to describe a field, as in bivector field, is more appropriately restricted to differential geometry? Should the name of this article be changed to p-vector field? Brews ohare ( talk) 16:55, 27 December 2009 (UTC) reply