What kind of "vector" is a feature vector in machine learning? I'm having trouble understanding the use of Vector in machine learning to represent a group of features.

If one looks up the definition of a Vector, then, according to wikipedia, a Vector is an entity with a magnitude and direction.

This can be understood when applying Vectors to for example physics to represent force, velocity, acceleration, etc...: the components of the Vector represent the components of the physical property along the axes in space. For example, the components of a velocity vector represent the velocity along the x, y and z axes

However, when applying Vectors to machine learning to represent features, then those features can be totally unrelated entities. They can have totally different units: one feature can be the length in meters of a person and another can be the age in years of the person.

But then what is the meaning of the Magnitude of such a Vector, which would then be formed by a summation of meters and years? And the Direction?

I do know about normalization of features to make them have similar ranges, but my question is more fundamental.
I'm having trouble understanding the use of Vector in machine learning
  to represent a group of features.


In short, I would say that "Features Vector" is just a convenient way to speak about a set of features. 


Indeed, for each label 'y' (to be predicted), you need a set of values 'X'. And a very convenient way of representing this is to put the values in a vector, such that when you consider multiple labels, you end up with a matrix, containing one row per label and one column per feature.

In an abstract way, you can definitely think of those vectors belonging to a multiple dimensions space, but (usually) not an Euclidean one. Hence all the math apply, only the interpretation differs !

Hope that helps you.