Dot Product Projection: What Does A Dot B Mean?

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SUMMARY

The discussion focuses on the dot product of vectors A and B, clarifying its geometric interpretation and applications in mathematics and science. The scalar projection of vector B onto vector A is defined as B multiplied by the unit vector of A or [A dot B]/[magnitude of A]. The dot product A.B is expressed as |A||B|cos(x), where x is the angle between the vectors, indicating its utility in calculating volumes and other geometric properties. The conversation emphasizes the importance of understanding the dot product beyond its computational aspect.

PREREQUISITES
  • Understanding of vector operations, specifically dot product and cross product
  • Familiarity with geometric interpretations of vectors
  • Knowledge of trigonometric functions, particularly cosine
  • Basic concepts of vector magnitude and unit vectors
NEXT STEPS
  • Explore the geometric interpretation of the dot product in vector calculus
  • Learn about the applications of the dot product in physics, particularly in calculating work
  • Investigate the properties of the cross product and its relationship to the dot product
  • Study the triple product and its significance in determining volumes
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Students and professionals in mathematics, physics, and engineering who seek to deepen their understanding of vector operations and their applications in various fields.

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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