What is the meaning of the dot product in calculus?

[steez]

I was woundering what exactly is the dot product and by that I mean what does it represent because I know the equations but it just seems to spit out a random number. I do not get what this number is supposed to mean. I know how it is usefull to solve many different problems and I know how to use it but it doesn't make very much sense because how is it related to anything. I have seen proofs but they don't seem to help. What does it acctually mean?Or am I thinking of the dot product in the wrong way?

 

[lurflurf]

The dot product generalizes the concepts of length and distance.
say
u,v,w are vectors
a is a scalar
||u||
||v||
are lengths
u.v=||u||*||v||*cos(angle)
u.u=||u||*||u||

dot product is linear
there are more general inner products, but it is usual to suppose the dot product has these properties

positivity
v.v>=0
definiteness
v.v=0 if and only iv v=0
additivity
(u+v).w=u.w+v.w
homogeneity
(av).w=a(v.w)
symmetry
u.v=v.u

also of great importance is invariance under isometry
let A be an isometry (ie a change in coordinates)
(Au).(Av)=u.v

dot product is the only (up to multiplication by a scalar) product map from two vectors to a scalar that is invariant

The meaning of a vector does not depend on its coordinates so any question about the relationship between two vectors does not either. Thus dot products are often used to answer such questions.

Dot products can also be used to project vectors.
That is pick out a relavent part.
to find components of a vector we use
u.i
u.j
u.k
where i,j,k is an orthonormal basis

 

[rochfor1]

I agree with lurflurf...an interesting interpretation of the dot product is the projection of one vector onto another. That is, if ||u||=1=||v||, then u.v is the component of u in the v direction (similarly the component of v in the u direction). Things get more complicated when we deal with non-unit vectors, because you have to normalize by ||u||^2 to get the correct projection.

 

[Nick M]

I like to think of it in terms of work/energy (although I realize there are any number of applications both practical and abstract).

Think of the current in a river as the force, and your movement across the river as the displacement. The dot product of the current and your displacement indicates the work done on you by the current. If you swim with the current, the angle between the current vector and your displacement vector is small (cosø is positive), and you don't have to exert yourself as much (if at all, ie. cos(0)=1). If you battle the current to swim upstream or go horizontally across to the other bank (forcing you to swim at an angle upstream) you have to expend more energy to battle the force of the current.

Or think of it as the "Shadow" of one vector cast on another using a light source perpendicular to the latter vector (a "projection").

Essentially it is the portion of one vector that acts in the direction of another.

Now how about the cross product...