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MathGroup Archive 1998

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Re: dot product and inner product?



s2700114@nickel.laurentian.ca wrote:
> 
> What is the difference between the commands dot product and inner
> product as  well as the commands cross product and outer product?
Inner[Times,x,y,Plus] is identical to the dot product.  It is simply
more general so you can use other functions besides Times and Plus.

Cross product and outer product have nothing to do with one another. 
Cross product is a vector cross product defined as summation of all
possible products between the elements of two vectors with even
permutations being multiplied by one and odd permutations being
multiplied by -1 and all others by zero.  In tensor notation, A_i B_j
epsilon_k.

Outer product makes a matrix that is the same size as the rows of the
first and the columns of the second(or vice versa).  If it is a row and
columnn vector, then the [i,j]th element of the outer product is equal
to the ith element of the row vector times the jth element of the
column vector.  Of cours, with the mathematica command Outer, you
aren't restricted to multiplication.

Notice that dot/Inner products produce a result that is one tensor rank
below their arguments while cross and outer products produce a result
that is one tensor rank above their arguments.  (a cross product is
really a degenerate rank two tensor, not a vector).  Hope that helps.

If you are really interested in this, a "mathematica methods
for..(physics, engineers) " kind of book will usually have a section
about group theory and tensor analysis which uses these kinds of
products extensively.  An introductory physics or mechanical
engineering book will have a section about vectors and dot/cross
products. -- 
Remove the _nospam_ in the return address to respond.



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