Are there different types of vectors? In linear algebra we have vectors:$$
\mathbf{A}=(x,y,z)=x\mathbf{\hat e}_x+y\mathbf{\hat e}_y+z\mathbf{\hat e}_z$$
We have vector algebra, i.e. vector addition, dot product, lines, planes, etc. A vector have a magnitude and a direction.
However, in multivariable calculus we also have vectors:$$
\mathbf{A}(t)=(x,y,z)=x\mathbf{\hat e}_x+y\mathbf{\hat e}_y+z\mathbf{\hat e}_z
$$
Here we do derivatives and integrals.
What is the difference?  Are there different types of vectors? 
I have always thought of vectors as the representation in the above link.

Multivariable calculus  is essentially the study of functions between vector spaces.  A function $f: \mathbb{R}^m \to \mathbb{R}^n$ is a function of $m$ variables that represents a field  of $n-$dimensional vectors.
