hmm.. taking a look at the "geometrical analysis" section of my "handbook of mathematics" (bronstein, semendjajew - dunno if this book exists in an english version) it seems that you´re right.
I still think the multilinear-form approach is better because it´s more general (I know this is a very subjective statment). But I can live with the idea of fixing the remaining degree of freedom (so that A scalar product becomes THE dot product) with a geometrical postulation/demand/claim (whatever the correct word is) - the cos-thing.
dot product and it's derivation
The neat thing about the |A||B|cosC definition of the dot product is that the equation does not change with different numbers of dimensions, as opposed to AxBx+AyBy and AxBx+AyBy+AzBz, etc. Nice static definition applicable to space of any number of dimensions.
Doc, I agree with you except for the fact that you still have to calculate the length of your A and B vectors - which requires the sum of componentwise multiplications anyway :D
Athiest, just to address one of your earlier points: it does seem kinda tricky defining the angle for vectors of more than 2 dimensions. However, it also seems that even the pythagorean length of vectors for more than 3 dimensions doesn''t make sense either - how would you factor (x,y,z,smell) into a length? Well, that''s maths for you - the theory comes first (several hundred years before at times), and then the application...
Maybe one day we will be able to rationalize things like the angle in 103 dimensional space and the length of a 45 dimensional vector, but till then, I guess I''ll just stick with what I know and what other people can teach me and just rely on the basis of maths.
Athiest, just to address one of your earlier points: it does seem kinda tricky defining the angle for vectors of more than 2 dimensions. However, it also seems that even the pythagorean length of vectors for more than 3 dimensions doesn''t make sense either - how would you factor (x,y,z,smell) into a length? Well, that''s maths for you - the theory comes first (several hundred years before at times), and then the application...
Maybe one day we will be able to rationalize things like the angle in 103 dimensional space and the length of a 45 dimensional vector, but till then, I guess I''ll just stick with what I know and what other people can teach me and just rely on the basis of maths.
>> how would you factor v=(x,y,z,smell) into a length?
with length(v) = ||v||. It´s called the norm and makes your vector space a normed vector space (that´s one where you can measure lenghts).
Maybe I should also clearify that by "more general" I wasn´t just referring to vector spaces R^n but to any vector spaces. Also don´t think that vector spaces other than euclidean R^n are science fictional. They are absolutely standard being used in differential geometry (theory of relativity) and in quantum mechanics (you´d have a hard time defining a geometrical angle between two gaussians).
btw.: You could start spelling my name correctly
Doc: Try < x | y > = x1y (1 being the unit matrix).
[edited by - Atheist on October 26, 2003 9:14:03 AM]
with length(v) = ||v||. It´s called the norm and makes your vector space a normed vector space (that´s one where you can measure lenghts).
Maybe I should also clearify that by "more general" I wasn´t just referring to vector spaces R^n but to any vector spaces. Also don´t think that vector spaces other than euclidean R^n are science fictional. They are absolutely standard being used in differential geometry (theory of relativity) and in quantum mechanics (you´d have a hard time defining a geometrical angle between two gaussians).
btw.: You could start spelling my name correctly
Doc: Try < x | y > = x1y (1 being the unit matrix).
[edited by - Atheist on October 26, 2003 9:14:03 AM]
Sorry for reviving this thread (1 week old) but I thought you should know:
Richard Feynman uses the formula:
axbx + ayby + azcz = a*b
when he defines the dot product. This is explained in volume 1 chapter 11 of his lectures.
The reason for which I think people say that dot product is |a||b|*cos(theta) is because of the way it was computed the first time. People obviously defined in the beginning scalar product like that (we can actually SEE what this means).
After this, when linear algebra was invented, people became interested in having a generalization of this stuff. The second formula was appliable in many more situations than the second formula (and was able to define the angle between two vectors in n-dimensional spaces). THe second formula (with the cosinus) became just a geometric interpretation of the limited case with 3 or 2 dimensions.
The cos formula for the dot product was important because it allowed a very important generalization, but the RIGHT definition is that without cos.
Richard Feynman uses the formula:
axbx + ayby + azcz = a*b
when he defines the dot product. This is explained in volume 1 chapter 11 of his lectures.
The reason for which I think people say that dot product is |a||b|*cos(theta) is because of the way it was computed the first time. People obviously defined in the beginning scalar product like that (we can actually SEE what this means).
After this, when linear algebra was invented, people became interested in having a generalization of this stuff. The second formula was appliable in many more situations than the second formula (and was able to define the angle between two vectors in n-dimensional spaces). THe second formula (with the cosinus) became just a geometric interpretation of the limited case with 3 or 2 dimensions.
The cos formula for the dot product was important because it allowed a very important generalization, but the RIGHT definition is that without cos.
quote:... but the RIGHT definition is that without cos.
That is simply not true. There is no one right definition. Those 2 definitions are equivalent, but in different terms. Which definition you should use depends upon the context of the problem.
The dot product itself is just the inner product of two 3D vectors.
[edited by - Mastaba on November 3, 2003 8:08:57 PM]
This topic is closed to new replies.
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