Force Vectors and Dot Products

[ewan_71]

Hello all ,

I am interested in the following problem:

In an ensemble of atoms, the forces on atom A and atom B are given by the vectors Fa=Fax+Fay+Faz and Fb=Fbx+Fby+Fbz, respectively.

Their respective positions are given by,

Pa=Pax+Pay+Paz and Pb=Pbx+Pby+Pbz

I have two questions:

(1) Would I be right in thinking that the dot product of Fa and Pb will give the magnitude of the force on atom A in the direction of atom B?

(2) Also, is the dot product of Fa and Fb the attractive force between the atoms?

Any help would be greatly appreciated !

Ewan.

 

[Doc Al]

I have two questions:

(1) Would I be right in thinking that the dot product of Fa and Pb will give the magnitude of the force on atom A in the direction of atom B?

No. To find the component of Fa in some direction, take its dot product with the unit vector in that direction. The direction from A to B will be parallel to Pb-Pa.

(2) Also, is the dot product of Fa and Fb the attractive force between the atoms?

No. (That quantity would not even have units of force.)

 

[Born2bwire]

No and no.

You would want the directional vector that points from A to B. So it would be:
$$\hat{R}_{ab} = \frac{\mathbf{P}_b-\mathbf{P}_a}{\left|\mathbf{P}_b-\mathbf{P}_a\right|}$$
Then you would do the dot product of Fa and \hat{R_{ab}}.

I assume by attractive force you just mean the magnitude of the force in the direction towards the other atom (regardless of whether or not the atom's are causing the attraction since you do not specify how these forces arise). The force on A towards B is just Fa\cdot\hat{R_{ab}}\hat{R_{ab}}. That is, it is the dot product of Fa and R_{ab} scaling the vector R_{ab}. Likewise, for B, you use Fb and -R_{ab}.

 

[ewan_71]

Thanks guys that's extremely helpful! :)