Answer to Question #81008 in Physics for DD

Question

Why is the product of two vectors a scalar quantity?

Answer 1

One might say because it is defined that way.
The dot product has the geometric interpretation as the length of the projection of A onto the unit vector B when the two vectors are placed so that their tails coincide.
In physics it represents a quantity that arises from vector quantities, but is a scalar in nature, hence coordinate independent. From example the work (scalar) done by a force (vector).