What is the meaning of Dot Product

[Shing]

I just reviewed Dot Product,
but I don't know what it actually, exactly means.
would you tell me about its physical meaning or something interesting quality of it?

Thanks

 

[Werg22]

Look it up in terms of projection of a vector on another...

 

[quasar987]

did u check wiki?

http://en.wikipedia.org/wiki/Dot_product

 

[Hurkyl]

I just reviewed Dot Product,
but I don't know what it actually, exactly means.
would you tell me about its physical meaning or something interesting quality of it?

Thanks

Well, a dot product can only possibly have a physical meaning when you're using it on vectors to which you've ascribed a physical meaning.

The dot product satisfies the commutative law, the distributive law, and it also satisfies an associative law with scalar multiplication -- those are very interesting qualities! And because the dot product is scalar valued, it allows you to reduce questions about vectors to questions about scalars! Because of these nice properties, the dot product is very convenient algebraic operation.

Geometrically, dot products are intimately related to lengths and angles. In fact, in many circumstances, dot products are used to define the notions of length and angle.

 

[Shing]

did u check wiki?

http://en.wikipedia.org/wiki/Dot_product

Yes, I did.
I just wonder what mathematician create Dot Product for.
I know the meaning of it as how book told me.
But I think my knowing is not enough.

Thank you

 

[Shing]

Like for example I always can't comprehend when they can separate a vector into two vectors. Or it just a definition? If so, why the mathematicians defined it?

 

[Hurkyl]

Like for example I always can't comprehend when they can separate a vector into two vectors.

You can always write a number as a sum of two numbers, right? What's so hard about the same for vectors?

 

[Shing]

You can always write a number as a sum of two numbers, right? What's so hard about the same for vectors?

I am sorry, I mean a vector can be separated into two vectors of different direction.

 

[Hurkyl]

I am sorry, I mean a vector can be separated into two vectors of different direction.

Let v be a vector you want to write this way. Let w be a vector pointing in a different direction. Let x=v-w. Then, we can write v=x+w.

Exercise: prove x and w point in different directions. (Hint: one way is to relate the notion of "different direction" to dot products)

 

[Shing]

Well, a dot product can only possibly have a physical meaning when you're using it on vectors to which you've ascribed a physical meaning.

The dot product satisfies the commutative law, the distributive law, and it also satisfies an associative law with scalar multiplication -- those are very interesting qualities! And because the dot product is scalar valued, it allows you to reduce questions about vectors to questions about scalars! Because of these nice properties, the dot product is very convenient algebraic operation.

Geometrically, dot products are intimately related to lengths and angles. In fact, in many circumstances, dot products are used to define the notions of length and angle.

I believe that is exhilarating.
But I am still in high school, not understanding why something satisfies the commutative law, the distributive law... and the other things don't.

 

[Shing]

Let v be a vector you want to write this way. Let w be a vector pointing in a different direction. Let x=v-w. Then, we can write v=x+w.

Exercise: prove x and w point in different directions. (Hint: one way is to relate the notion of "different direction" to dot products)

May I know why we can use Binary operation on vector? Thanks!

 

[david07]

dot product is a.b = |a||b|cosQ where Q is the angle

Also if two vectors are considered orthogonal if their dot product is zero (a.b=0)

vector product is a*b = |a||b|sinQn where n is a unit vector perpendicular to the plane containing a and b.

while in a cross product if two non-zero vectors a and b are parallel it is a × b = 0.

 

[prasannapakkiam]

Well here are the following uses:

1. Calculating Work (Cross Product for Torque), W=f.d - force and distance are given as vectors. Notice how, the dot product eliminates all components of the force vector that arn't in the same direction as the distance vector...

2. As David said, it can be used to work out the angle between 2 vectors.

3. As I used to tell people: The Dot Product measures the 'Right-Angularity' of 2 vectors.

4. Dot Products; since they take the components only in the same direction, they can be useful when doing projections.