In more general terms, the product of two vectors is a scalar.
Take, for instance, the calculation of work.
W = F x d
W = Work (scalar)
F = Force (vector)
d = displacement (vector)
Mathematically this occurs because you are taking the dot product of the two vectors. In three-dimensional space, the coordinate directions are represented by i, j, and k. The dot products are as follows for the unit directions:
i (dot) i = 1
i (dot) j = 0
i (dot) k = 0
and so on. Example:
F = 3i + 4j + 2k
d = i + 2j + 7k
W = (F) dot (d) = (3i + 4j + 2k) dot (i + 2j + 7k)
Using normal distributive property and you get
W = (3i)(i) + (4j)(2j) + (2k)(7k) = 3 + 8 + 14 = 25
Remember any terms with ij, ik, ji, jk, ki, or kj become ZERO by definition of a vector product.
Bomba is very confused indeed!
"The product called work is a vector as is momentum."
Work is a scalar because it has only magnitude. Momentum is a vector (has magnitude and direction) and is the product of a scalar (mass) and a vector (velocity).
"The multiplication of F and D and of M and V both result in a larger scalar value, but does not change the direction of D or of V."
It's not really "multiplication", but rather the dot product as I explained. Alternatively, if you take the cross product of two vectors, like force and distance, you get an entirely different quantity; Torque. Which is, of course, a vector.
"However, when different vectors are combined the result will be a different scalar value and a different direction."
"Combined" in what fashion? It is impossible for a scalar to have direction!
"This is seen in the calculation of kinetic energy where the velocity vector is squared and the resulting energy is still a vector."
Absolutely false! Energy is a scalar! In simple mechanics, the total energy of an object is an algebraic sum (rather than a vector sum):
U = KE + PE
Many years of teaching college-level statics and dynamics