Geometrical interpretation of dot product?

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Discussion Overview

The discussion centers around the geometrical interpretation of the dot product in three dimensions, particularly in relation to calculating the dot product between two vectors and understanding its implications in a Cartesian coordinate system. Participants explore the transition from a cosine-based interpretation to a more geometric approach, considering the integration over angles and orientations of vectors.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether the 2D interpretation of the dot product can be extended to 3D by adding an additional cosine term.
  • Another participant clarifies that the dot product in 3D remains |a||b|cos(theta), where theta is the angle between the two vectors, and relates the dot product to the projection of one vector onto another.
  • A participant expresses uncertainty about how to transform the cosine(theta) notation into a Cartesian coordinate system, noting that the resulting value of the dot product depends on the angle between the vectors.
  • One participant requests a specific example to better understand the problem being discussed.
  • A later reply provides a specific formula involving transition dipoles and a vector, indicating a need to integrate over all orientations of the dipoles, while expressing the challenge of moving from 2D to 3D integration.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to interpret the dot product in 3D or how to effectively transition from a cosine-based interpretation to a Cartesian framework. Multiple views on the interpretation and application of the dot product remain present.

Contextual Notes

Participants express uncertainty regarding the transformation of angles and the implications of integrating over orientations in three dimensions, highlighting potential complexities in the Cartesian coordinate system.

excalibur313
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Does anyone know what the geometrical interpretation of a dot product in 3-D is? I am calculating the dot product between two vectors in 3d and need to use the |a||b|cos(theta) interpretation basically, but that is for 2d. Can I just tack on an additional cos(theta2)? Thanks a lot for your help!
-Stephen
 
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excalibur313 said:
Does anyone know what the geometrical interpretation of a dot product in 3-D is? I am calculating the dot product between two vectors in 3d and need to use the |a||b|cos(theta) interpretation basically, but that is for 2d. Can I just tack on an additional cos(theta2)? Thanks a lot for your help!
-Stephen

Nope... it's still |a||b|cos(theta) where theta is the (smaller) angle between the two vectors. If you have the rectangular components of the vectors, you can use
a_x b_x + a_y b_y + a_z b_z (which you might which to prove).

The usual geometrical interpretation of the dot-product is related to a projection of one vector onto the other.
 
Thanks a lot for your help, but I guess I am wondering what the best way to approach this problemis. The idea is that the resulting value of this dot product is based on the angle between them. (Where it will be a bigger value if they are lined up parallel) so it isn't really apparent to me how best to transform from cosine(theta) notation to cartesian coordinate system. Yeah, sorry I'm not trying to turn this into a homework help session, but basically I want to integrate over all possible angles so a cartesian coordinate system would be messy I think.
 
Last edited:
Can you give a specific example... what do you have to start with?
 
Okay, well the rate of fret is proportional to K^2 where:
K=3(p1.r)(p2.r)-(p1.p2)
Where p1 and p2 are the transition dipoles and r is a vector pointing from the center of p1 to the center of p2. r remains constant so I have to find <k^2> so I basically want to integrate over all orientations of the two transition dipoles. If this was in 2d I could just integrate over theta 1 and 2 because the angle between p1 and p2 is just the difference between those two. It looks like this: (p1) | ------> | (p2) where the arrow is r and the two vertical lines are the transition dipoles and can rotate. Anyway, thank you so much for anything you can do.
 

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