help with understanding vector functions. I was reading a calculus book in which there was a definition of vector function as: A function which assigns to every real number in the domain of a vector function, a vector in V(3). But i don't know what V(3) means. Please help.
 A: That is pretty non-standard notation, but V(3) means 3-dimensional euclidean space. It is the space of tuples (x,y,z).
A: It seems that the definition refers to a function that has as independent variable a real number and has as value a three dimensional vector. I.e. a function: $F:\mathbb{R}\rightarrow \mathbb{R}^3$. A function of this type has form:
$$
\begin{bmatrix}
y_1\\
y_2\\
y_3
\end{bmatrix}
=F(x)=
\begin{bmatrix}
F_1(x)\\
F_2(x)\\
F_3(x)
\end{bmatrix}
$$ 
with $F_i:\mathbb{R}\rightarrow \mathbb{R}$ are real functions.

You can think at the vector 
$$
\begin{bmatrix}
y_1\\
y_2\\
y_3
\end{bmatrix}$$ 
simply as a point in $3-$dimensional space and a simple example of such a function is a rule that defines the coordinates of a point  $P=(x,y,z)$ in such a way that:
$$
x=1+2t \qquad y=2-t \qquad z=1+t
$$
where $t$ is a real number. All the points given by this function stay on a stright line.
A: It's the 3D Euclidean space. The space we live in and are used to.
A good example is the temperature of each point in the room, let's represent this function with  T(x,y,z). For each coordinate (x,y,z) the function T(x,y,z) gives you the temperature inside the room.
