What does this mean (vectors)

A function of a vector, say f(r) , is called linear if

F(r1 + r2) = f(r1) + f(r2), and f(ar) = af(r)


what is this?

It means that a vector-valued function is called liner if the function of the sum of the two functions is equal to the sum of the functions, and the value of the function multiplied by a constant a is equal to the function of the argument multiplied by constant a.

they are the properties that define a linear function.

the easiest way to understand them is via example.

f(r) = r.r = r^2 is not a linear function because

f(r1) = r1^2
f(r2) = r2^2
f(r1+r2) = (r1+r2)^2 = r1^2 + 2 r1.r2 + r2^2 = f(r1) + f(r2) + 2 r1.r2
so
f(r1+r2) != f(r1) + f(r2)

likewise, f(ar) = (ar)^2 = a^2 r^2 = a^2 f(r)
so
f(ar) != a f(r)

now do the same analysis for f(r) = r [which is known to be linear] and see that the two properties are satified.