Wikipedia:Reference desk/Archives/Mathematics/2011 November 5

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November 5

Cross vs Wedge product

Hello. I am confused about the difference between these two vector operations. I was taught that the magnitude of the cross product of two vectors is the area of the parallelogram they define but I have read that the wedge product is actually more associated with the area. I have tried your articles to elucidate but unfortunately they are written at a level above my understanding. Can someone please just clarify to me what the difference is? I know what the cross product is but I have never been taught the wedge product, which is probably why; just a simple explanation would go far in un-confusing me. Thanks profusely in advance. --Jacob — Preceding unsigned comment added by 68.1.61.47 (talk) 00:33, 5 November 2011 (UTC)

In three dimensions they can almost be thought of as the same thing (though professional mathematicians might have me shot for saying so!) The wedge product is actually a bivector whereas the cross or vector product is an ordinary perpendicular vector, but they both have the same magnitude (the area of your parallelogram). A fairly simple explanation can be found at Cross_product#Wedge_product. Dbfirs 00:53, 5 November 2011 (UTC)

Be careful about the dimensions. The cross product is only defined for vectors in 3-dimensional space. If you've heard someone mention the "cross product" of two vectors in the plane then they really mean the determinant of the matrix whose first row is the the first vector and whose second row is the second vector. For the area of the parallelogram spanned by two such vectors you calculate the absolute value of this determinant, i.e.

$(1,2)\times (3,4)=\left|\det \left[{\begin{array}{cc}1&2\\3&4\end{array}}\right]\right|=2\,.$

In terms of volume, this definition can be extended. If you have n vectors in n-dimensional space then the volume of the n-dimensional analogue of a parallelepiped is the absolute value of the determinant of the n-by-n matrix whose first row is the first vector, whose second row is the second vector, and so on.

If you have (n–1)-vectors in n-dimensional space and you want a vector that is perpendicular to the others then you can use the following trick. Use n so-called dummy variables, say x₁, x₂, …, x_n, and put them as the first row of an n-by-n matrix. Then put your (n–1)-vectors as the remaining rows in the n-by-n matrix. Finally, calculate the determinant; the coefficient of x₁ will be the first component of the perpendicular vector, the coefficient of x₂ will be the second component, and so on. For example, if the you had the vectors (1,0,1,0), (0,1,0,1) and (1,2,3,4) in 4-dimensional space then you'd calculate

$\det \left[{\begin{array}{cccc}x_{1}&x_{2}&x_{3}&x_{4}\\1&0&1&0\\0&1&0&1\\1&2&3&4\end{array}}\right]=-2x_{1}+2x_{2}+2x_{3}-2x_{4}\,,$

meaning that (–2,2,2,–2) is perpendicular to (1,0,1,0), (0,1,0,1) and (1,2,3,4). — Fly by Night (talk) 12:31, 5 November 2011 (UTC)

Geometrically, the wedge product of two vectors is the family of parallelograms with the same area lying in the same plane with the same orientation of the boundary (see image). Thus the wedge product has three pieces of information: the area of the parallelogram, the plane containing the parallelogram, and the orientation of the parallelogram. The cross product is the normal vector to that plane, whose magnitude is that area, and whose direction is determined by the right hand rule. This also contains the same information, but is only available in three dimensions and is thought of slightly differently (as a vector rather than as a parallelogram). Sławomir Biały (talk) 12:47, 5 November 2011 (UTC)

Interpretation of 'pullback of a pullback'

Hello everyone, I'm attempting some homework on a Category theory problems sheet and am having trouble interpreting the last part of a question, was hoping you could help.

The first 2 parts of the problem (which i've done) are about a concatenation of 2 pullback squares (http://www.dpmms.cam.ac.uk/~jg352/pdf/CTSheet2-2011.pdf Q2): the third part of the problem asks you to deduce, from the fact that combining 2 of the pullbacks in the appropriate way (e.g. concatenating the 2 small squares) gives you another pullback, that "the pullback of a pullback square is a pullback". Now this is where I'm confused; what is "the pullback of a pullback square"? To me, a pullback is the limit of a diagram consisting of 2 morphisms with the same codomain (as defined on the wikipedia article). So then, how can we take a pullback of a pullback square? The only thing I could think of would be to take the pullback of the 2 maps which our square tells us commute (i.e. in pullback, these would be $gp_{2},\,fp_{1}$): but then if we are simply taking a 'pullback of a pullback square' to be the pullback of these 2 morphisms, then how could it possibly not be a pullback?

I get the impression from the question that I'm meant to be looking at a cube and combining various combinations of faces with the first 2 parts of the problem to show that lots of other things are also pullbacks, but I don't think I even know what I'm trying to show. I guess the 'pullback of a pullback square' is not meant to be assumed to be a pullback of the 2 commuting maps, otherwise the question would be done. Could anyone help? Does anyone know what the question actually requires me to show? (And whether fiddling about with faces on a cube is the right way to go about it?) Very confused, would be very appreciative of any help you can offer: thank you! 86.26.13.2 (talk) 22:05, 5 November 2011 (UTC)

Obverse in symbolic logic

What is the obverse of Q -> P in symbolic logic ? — Preceding unsigned comment added by 76.67.165.103 (talk) 22:17, 5 November 2011 (UTC)

I'm not familiar with obverse, but a peek at our page on it makes it sound like it's just De Morgan's law by another name. So maybe $\neg (Q\wedge \neg P)$. Here I've translated implication into a disjunction, applied De Morgan's law, then eliminated double negation.--121.74.125.249 (talk) 22:37, 5 November 2011 (UTC)

Our article is obversion. I was not familiar with it either, but from the article it seems to be related to contraposition, except that it applies only to categorical propositions. The contrapositive of Q -> P is ~Q -> ~P (as 121.74 stated), but as Q -> P is not a categorical proposition, I'm not sure that is has an obverse. -- ToE^T 12:21, 10 November 2011 (UTC)

Polish notation

How would this go in Polish notation ? ¬(¬q v ¬p) — Preceding unsigned comment added by 76.67.165.103 (talk) 22:47, 5 November 2011 (UTC)

¬v¬q¬p is it in Polish and q¬p¬v¬ is reverse Polish. See Polish notation Dmcq (talk) 23:33, 5 November 2011 (UTC)

Just realized you might mean with the original logical notation. That's a fairly easy one fo one swap of letters for the symbols. Dmcq (talk) 23:36, 5 November 2011 (UTC)

Is it NANqNp in original notation ? I know, I'm not an expert, but please correct me anyway. — Preceding unsigned comment added by 76.67.165.103 (talk) 23:39, 5 November 2011 (UTC)

Yes. Dmcq (talk) 23:46, 5 November 2011 (UTC)