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I'm not sure I understand your question well.
My answer would simply be: Because these operations were built as such by the mathematicians.
Physicists use them to model different quantities that also happen to be scalar or vector.
You will choose dot- or cross- product according to the quantity you want to express.
The work done by a force will be the result of a dot-product (this is a consequence of the definition of work) , whereas Laplace's force, applied by a magnetic field on a conducting wire will be defined by a cross-product.
ooooooook now i got it. Thanks for your time sir. Thank you very much. God bless you.
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the dot product indicates only the product of the magnitudes(no direction) but the cross products indicate the product of the vectors(with direction) The dot product is used when the direction is universal. I.e. there is only one direction. The cross product is used when the direction is not universal. i.e. there is one or more directions it can result to. The dot product and cross product are there to prevent confusion.
thats what i was just going to say...,but you already said it shadowys...
thats the concept ahead of maths...
A dot product is an "inner" product, roughly speaking, it measures what two vectors have in common. Consequently it is a simpler object than either vector. A cross product is an "outer" product, roughly speaking, it measure the range of other vectors you can create using bits from both vectors. Consequently it is a more complex object. A still more complex object is a "direct" product, which would be the range of any kind of object you can make from two vectors. It's a tensor, and has 9 components.