Dot Product Projection: What Does A Dot B Mean?

[mvpshaq32]

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?

 

[mfb]

What do you mean with "meaning"?
Geometric interpretation? That is tricky.
Application in mathematics and science? It is very useful for many things.

 

[mathman]

A.B = |A||B|cosx, where x is the angle between the vectors.

 

[mvpshaq32]

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?

 

[mfb]

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.