Abuse of Notation - Cross Product

[prasannaworld]

http://en.wikipedia.org/wiki/Abuse_of_notation

I don't want this thread to go too long. But can someone quickly explain to be how the Determinent method of evaluating the Cross Product is an "abuse". I cannot quite seem to grasp their explanation...

 

[Office_Shredder]

Basically, a matrix is defined to have a scalar (usually a real or complex number) in each entry, but when using the cross product calculation, you plug a vector into each entry of the top row. So you really don't have a matrix at all, but you still plug it into the determinant function and treat it like one because it's convenient

 

[D H]

Cool! I thought I was the only one who objected to that silly formulation on the basis that it is an incredible abuse of notation. Thanks for the pointer.

That it happens to work is no excuse. It is abuse of notation. You have to "forget" that $\hat{\boldsymbol{i}}, \hat{\boldsymbol{j}}, \hat{\boldsymbol{k}}$ are vectors when you put them in that matrix, the "remember" that they are vectors after computing the deteriminant (which is by definition a scalar).

 

[prasannaworld]

ok...

I kept thinking of the Im(a*b) definition... Forgot completely about what exactly the Determinent is... Thanks for reminding me

 

[HallsofIvy]

I remember in one class, a student who was presenting a proof in class, saying "by abuse of notation, ..." and the professor immediately saying "Let's not be that abusive"!

 

[Office_Shredder]

Cool! I thought I was the only one who objected to that silly formulation on the basis that it is an incredible abuse of notation. Thanks for the pointer.

That it happens to work is no excuse. It is abuse of notation. You have to "forget" that $\hat{\boldsymbol{i}}, \hat{\boldsymbol{j}}, \hat{\boldsymbol{k}}$ are vectors when you put them in that matrix, the "remember" that they are vectors after computing the deteriminant (which is by definition a scalar).

It's not a definition, it's more like a mnemonic. It's like complaining that SOHCAHTOA isn't a real number so it's not a valid way of remembering what cosine, sine and tangent are.

 

[Tac-Tics]

Most abuse of notation is easy to spot because it doesn't type check.

A matrix is a mathematical object built on top of a field (usually taken to be the reals or the complex numbers). For each row-column index that's within the bounds of the matrix, there is an entry which is a member of that field. Clearly, when you find a vector in row 1 and reals in all the other rows, what you have isn't a matrix of real numbers. What you have is a physicist or engineer =-P

It's not a definition, it's more like a mnemonic. It's like complaining that SOHCAHTOA isn't a real number so it's not a valid way of remembering what cosine, sine and tangent are.

I don't think anyone ever complained SOHCAHTOA isn't a real number. I always just complained it was hard to spell.

But the problem is, it often ISN'T explained that the notation is just a shorthand. You can see this even in Feynman's Lectures, where he introduces vector calculus operations grad, div, and curl as what happens when you apply a vector $$(\frac{\partial}{\partial x}}, \frac{\partial}{\partial y}}, \frac{\partial}{\partial z}})$$ to a function using scalar multiplication, dot product, and cross product. But in reality, applying a derivative is done with function application, and not multiplication!

 

[lurflurf]

One can define matrices over a ring and determinants of such matrices.
the problems are:
-this construction is often glossed over
-if the ring one defines is noncommutative determinant notation is confusing
if R contains i and j than it contains ij and ji, but do we consider thhem equal?
-the ring might contain objects one has immidiate use for
what use have we of iiiii?
-this construction is distracting and inelegant
-a better reminder would be index notation
axb=SUM[over i and j from 1 to 3]alternator(i,j,k)a(j)b(k)

 

[Crosson]

I really don't consider the use of the determinant for the vector cross product to be an abuse of notation. In some higher algebra contexts the determinant is really just a formal operation ie. the elements could be infinite dimensional operators. To say that some elements of the matrix are not numbers, as you think they should be, is to miss the entire point of a formula, viz that it is formally true in terms of symbols. The determinant merely indicates that we apply a certain formula.

I think many of the examples on the wikipedia page are not very well thought out. It has always seemed that "abuse of notation" is an oft-repeated but ill understood phrase, and this thread has only affirmed this impression.