Dot Product Projection: What Does A Dot B Mean?

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Discussion Overview

The discussion centers around the interpretation and meaning of the dot product of two vectors, specifically in the context of scalar projection and geometric significance. Participants explore various aspects of the dot product, including its applications in mathematics and science.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the meaning of the dot product A · B, particularly in relation to the scalar projection of vector B onto vector A.
  • Another participant prompts for clarification on what is meant by "meaning," suggesting that geometric interpretation is complex.
  • A participant states the mathematical relationship A · B = |A||B|cos(x), where x is the angle between the vectors.
  • There is a discussion about the geometric meaning of the dot product and what its value represents, comparing it to the cross product's representation of area.
  • One participant mentions that the dot product can be used to calculate volumes, particularly in the context of the triple product involving orthogonal vectors.

Areas of Agreement / Disagreement

The discussion reflects multiple competing views on the interpretation of the dot product, with no consensus reached on its geometric meaning or representation.

Contextual Notes

Participants express uncertainty about the geometric interpretation of the dot product and its significance compared to other vector operations, such as the cross product.

Who May Find This Useful

This discussion may be of interest to students and professionals in mathematics, physics, and engineering who are exploring vector operations and their interpretations.

mvpshaq32
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Simple question, but I don't know why I never learned this before.

If the scalar projection of vector B onto vector A is B * Unit vector of A (or [A dot B]/[magnitude of A]), then what does the dot product of simply A and B give you, assuming neither is a unit vector.

If it's not clear what I'm asking, it's that the component of vector B projected onto vector A is given by [A dot B]/[magnitude of A], but then what is the meaning of simply A dot B?
 
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What do you mean with "meaning"?
Geometric interpretation? That is tricky.
Application in mathematics and science? It is very useful for many things.
 
A.B = |A||B|cosx, where x is the angle between the vectors.
 
mfb said:
What do you mean with "meaning"?
Geometric interpretation? That is tricky.
Application in mathematics and science? It is very useful for many things.

Yes, exactly, the geometric meaning.

What does its value represent?

For example the magnitude of the cross product represents the area of the parallelogram formed by two vectors.

So what does the dot product represent?
 
You can use it to calculate volumes, for example, if you have the area of the floor given as (orthogonal) vector. This is used in the triple product.
 

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