The introduction of a scalar product achieves both of these goals.
British Dictionary definitions for scalar product
scalar product
noun
1.
the product of two vectors to form a scalar, whose value is the product of the magnitudes of the vectors and the cosine of the angle between them. Written: AยทB or ABAlso called dot product Compare vector product
scalar product in Science
scalar product The numerical product of the lengths of two vectors and the cosine of the angle between them.