- #1

murshid_islam

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thanks in advance.

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- Thread starter murshid_islam
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- #1

murshid_islam

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thanks in advance.

- #2

CompuChip

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Suppose I fix an origin in space (e.g. the center of the room) and consider a vector

Similarly, suppose that I have two vectors

This is probably the most clear use of the inner and cross product, though they (and their generalizations) turn up all over the place if you do math.

- #3

Hurkyl

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The English word "logic" is somewhat ambiguous. You are correct if you take it literally. However, your friends probably didn't mean it literally, and rather meant something like "Why would anyone have chosen to define that?"i think that there are no mathematical logic or explanations behind the definitions. they are just defined like that.

- #4

murshid_islam

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no my cousin meant "why are the definitions of vector and scalar product defined like that? what is the mathematical explanation behind why they are defined like that?"your friends probably didn't mean it literally, and rather meant something like "Why would anyone have chosen to define that?"

is there really any particular reason why they are defined like that? or did the mathematicians just chose to define them that way?

- #5

rcgldr

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Apparently someone decided that "scalar" needed to have more than one meanging, although there are other terms that avoid multiple meanings for "scalar".

Scalar normally means a dimensionless number, for example kinetic energy is a scalar.

You can find all of this information at wikipedia.

**Scalar**

**Scalar_multiplication**

**Dot_product or inner product**

The programming language APL extends the meaning of the inner (dot) product so it could be used to multiply matrices or any size variable with any other size variable, as long as the last dimension of the left variable has the same number of elements as the first dimension of the right variable.

**Matrix_multiplication**

Outer product and cross product not the same thing.

**Outer_product**

**Cross_product**

Yet another inner product:

**Scalar_product**

Scalar normally means a dimensionless number, for example kinetic energy is a scalar.

You can find all of this information at wikipedia.

The programming language APL extends the meaning of the inner (dot) product so it could be used to multiply matrices or any size variable with any other size variable, as long as the last dimension of the left variable has the same number of elements as the first dimension of the right variable.

Outer product and cross product not the same thing.

Yet another inner product:

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- #6

CompuChip

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no my cousin meant "why are the definitions of vector and scalar product defined like that? what is the mathematical explanation behind why they are defined like that?"

is there really any particular reason why they are defined like that? or did the mathematicians just chose to define them that way?

I don't know if my post gave the actual reason they are defined like that, but it seems at least like a very good one to me.

Also note that the scalar product picks out the component of one vector along the other, and the cross product picks out the perpendicular one. So if you have two forces and you want to factor one in a component parallel and a component perpendicular to the other, taking the inner and cross product will do that for you.

Again, as I already said, the scalar and cross products are specific instances of much more general concepts (namely, as Jeff Reid pointed out, inner and exterior (outer) products, respectively). So one could ask, why are those defined as they are, and is it a coincidence that these are special instances of them? (And the answer to the second question would then of course no)

- #7

murshid_islam

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- #8

danny271828

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- #9

radou

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Study linear algebra, and you'll have your answer.

The point is, an inner product space is a powerful ambient to work in. Here's just one example. If you take some vector space and some vector**v** in it, and, of course, if you chose a basis {**b1**, ..., **bn**} for the space, then there exist unique scalars v1, ..., vn such that **v** = v1 **b1** + ... + vn **bn**. But, one needs to *find* these scalars, right?

Well, in an inner product space (i.e. a vector space with an inner product defined on it), these scalars are known apriori, i.e. one can prove that for this same vector**v**, you have **v** = (**v** | **b1**) **b1** + ... + (**v** | **bn**) **bn**. (Where (.|.) denotes the scalar product.) Neat, huh?

Edit: CompuChip's post reminded me of the important fact which I forgot to point out, eg that {**b1**, ... ,**bn**} is an orthonormal basis.

The point is, an inner product space is a powerful ambient to work in. Here's just one example. If you take some vector space and some vector

Well, in an inner product space (i.e. a vector space with an inner product defined on it), these scalars are known apriori, i.e. one can prove that for this same vector

Edit: CompuChip's post reminded me of the important fact which I forgot to point out, eg that {

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- #10

CompuChip

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To make it what danny said more precise: suppose you have an object which is on a flat surface. Lay the

To get back to your example: let me give it a "practical" use. suppose I have a coordinate system in which

Hope that makes it more clear (not more confusing).

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