All the vector spaces considered for a product or quotient must be over the same type of $\field$ ($\real$ or $\comp$).
Suppose we have $m$ vector spaces $V_{1}, V_{2}, \ldots, V_{m} \in \field$, then their vector product \begin{align} V_{1}\times \cdots \times V_{m} = { (v_{1}, \ldots, v_{m}) : v_{1}\in V_{1}, \ldots v_{m} \in V_{m} }\end{align} Addition and scalarm multiplication are also defined on such products \begin{align} (u_{1}, \ldots, u_{m}) + (v_{1}, \ldots, v_{m}) &= (u_{1} + v_{1}, \ldots, u_{m} + v_{m})\newline \lambda (v_{1}, \ldots, v_{m}) &= (\lambda v_{1}, \ldots, \lambda v_{m})\end{align} where $\lambda \in \field$. With the above definitions of addition and scalar multiplication, the product of vector spaces is itself a vector space.
$\real^{2} \times \real^{3}$ is an example of a product of vector spaces. Elements of such a product are lists of the form $((x_{1}, x_{2}), (x_{3}, x_{4}, x_{5}))$ where all 5 elements belong in $\real$. Note that this product is technically different from $\real^{5}$, but is a simple relabelling of $\real^{5}$.
Dimension of a product of finite dimensional vector spaces is simply the sum of dimensions of the individual vector spaces \begin{align} \text{dim}(V_{1} \times \cdots \times V_{m}) = \text{dim }V_{1} + \cdots + \text{ dim }V_{m}\end{align} and can be shown by considering the bases vectors on either side.
We first define the meaning of adding a vector and a subspace. Suppose $v \in \setv$ and $U$ is a subspace of $\setv$. Then \begin{align} v + U = { v + u : u \in U }\end{align} For instance, consider $U = {(x, 3x) \in \real^{2} : x \in \real }$ and $v = (2,3)$. Then $v + U$ is the line with slope $3$ containing the point $(2,3)$.
Affine Subset and Parallel
An affine subset of $\setv$ is a subspace of $\setv$ of the form $v + U$ where $v \in \setv$ and $U$ is a subspace of $\setv$. Such a subset is said to be parallel to $U$.
Clearly, in the previous example, the subset ${(x + 2, 3x + 3) : x \in \field }$ is parallel to ${(x, 3x) : x \in \field }$ both by definition and physically (they are parallel lines).
Quotient Space $\boldsymbol{\setv/U}$
For any subspace $U$ of $\setv$, the quotient space $\boldsymbol{\setv/U}$ is the set of all affine subsets of $\setv$ parallel to $U$. \begin{align} \setv/U &= { v + U : v \in \setv } \text{dim } \setv/U &= \text{ dim } \setv - \text{ dim } U\end{align}
Two affine subsets parrallel to $U$ are equal or disjoint. Suppose $v, w \in \setv$ and $U$ is a subspace of $\setv$, then the following three statements will be equivalent
$v - w \in U$
$v + U = w + U$
$(v + U) \cap (w + U) \neq \phi$
Addition and Scalar Multiplication
Addition and scalar multiplication on $\setv/U$ are defined as follows \begin{align} (v + U) + (w + U) &= (v + w) + U\newline \lambda(v + U) &= (\lambda v) + U\end{align} With these definitions, $\setv/U$ is also a vector space.
Quotient Map $\boldsymbol{\pi}$ Suppose $U$ is a subspace of $\setv$, then the quotient map is a linear map from $\setv$ to $\setv/U$ defined as \begin{align} \pi = { v + U : v \in \setv }\end{align}