Dot Product: what does it symbolize?

Dark Knight 496

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Oct 8, 2005
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Ok, I understand that the dot product of vectors is the sum of the products of their corresponding elements, and that the dot product equals to the magnitude of the vectors time the cosine of the angle between them. What I do not understand is... what is the significance of dot product? how is it useful? what exactly does it symbolize? why the sum of the products of the elements?

please explain as simply as possible (i am in high school pre calc)... thanks...
 
You are in secondary mathematics, so I will try to give you a satisfactory answer to a very difficult question (it is really a good question). Vectors actually have more to do with being a scientific idea more than a mathematical notion. Vectors have both direction and magnitude those are not really mathematical properties. In general, objects in mathematics are sets. Therefore, many of the methods in vector analysis are not strictly mathematical with set theoretic basis: 'Dot Product' is one of those. Dot products have extreme utility just as you noted: angles between vectors; lengths of vectors; in probability theory; etc. Its mathematical ontology is somewhat doubtful. Its utility is absolutely essential in applied mathematics. I hope this makes some sense to you.
 
pka said:
You are in secondary mathematics, so I will try to give you a satisfactory answer to a very difficult question (it is really a good question). Vectors actually have more to do with being a scientific idea more than a mathematical notion. Vectors have both direction and magnitude those are not really mathematical properties. In general, objects in mathematics are sets. Therefore, many of the methods in vector analysis are not strictly mathematical with set theoretic basis: 'Dot Product' is one of those. Dot products have extreme utility just as you noted: angles between vectors; lengths of vectors; in probability theory; etc. Its mathematical ontology is somewhat doubtful. Its utility is absolutely essential in applied mathematics. I hope this makes some sense to you.

Thanks, but that does not really make much sense because it does not answer the primary question: what exactly is achieved by adding the products of corresponding elements?.. forget that, what exactly is a product of corresponding elements of two different vectors?
 
In mechanics, the dot product of a force vector with a displacement vector is the scalar value, work (energy).
 
Dark Knight 496 said:
what exactly is achieved by adding the products of corresponding elements?.. forget that, what exactly is a product of corresponding elements of two different vectors?
Because vectors are not really mathematical objects (i.e. they are not defined in terms of sets), there is no mathematical answer to that question. There are ‘scientific answers’. Look at the reply above.
 
Dark Knight 496 said:
Thanks, but that does not really make much sense because it does not answer the primary question: what exactly is achieved by adding the products of corresponding elements?.. forget that, what exactly is a product of corresponding elements of two different vectors?

From a geometric standpoint, taking the 'dot product' means multiplying "some part" of a quantity to a whole quanity. Take two vectors U and V. By taking their dot product U∙V, you are finding the value for some part of the length of U times the entire length of V.

This length is the length of the "shadow" of U on V times the length of V i.e. the length of U projected onto V times the length of V. That length is |U|*cos(theta) * |V|. You can verify for yourself that the component method will give the same answer. Should be a fun project.
 
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