why is $ \vec u \bullet \vec v = u_1v_1 + u_2v_2$? Let $ \vec u = \begin{bmatrix} u_1\\ u_2 \end{bmatrix}$ and $\vec v =  \begin{bmatrix} v_1\\ v_2  \end{bmatrix}$
Doesn't the number of columns in u have to equal number rows in v?
Why are they multiplied directly across?
EDIT: I feel that my question is unique to the one proposed (The dot product of two vectors.) b/c it succinctly addresses confusion arising from multiplication of two column-vectors, and the definition of $\vec u \cdot \vec v$.  
 A: The definition of the dot product of two vectors $u=(u_{1},u_{2})$ and $v=(v_{1},v_{2})$ is $u \cdot v = u_{1}v_{1}+u_{2}v_{2}$. We are not viewing these vectors as matrices when we multiply them (as you say, this makes no sense). However, we can view the dot product as the matrix product $u^{T}v$, where $u^{T}$ is a $1 \times 2$ matrix and $v $ is an $2 \times 1$ matrix. Their product is then a $1 \times 1$ matrix:
$$\pmatrix{u_{1} & u_{2}}\pmatrix{v_{1}\\v_{2}}=u_{1}v_{1}+u_{2}v_{2}$$
where, as usual, we view $1 \times 1$ matrices as scalars and drop the brackets.
A: The short answer is:
$$
\vec{u}\cdot\vec{v} = \vec{u}^TG\vec{v},
$$
where $^T$ is transpose and $G$ is a special matrix called metric tensor. For standard $\mathbb{R}^n$ space this tensor has signature $(n,0,0)$ which is just identity $n\times n$ matrix. Thus in $\mathbb{R}^2$
$$
\begin{pmatrix}u_1 \\ u_2\end{pmatrix}\cdot\begin{pmatrix}v_1 \\ v_2\end{pmatrix} = \begin{pmatrix}u_1 & u_2\end{pmatrix}\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}\begin{pmatrix}v_1 \\ v_2\end{pmatrix} = u_1v_1 + u_2v_2.
$$

In other spaces with other metrices the scalar product may has another forms. For example, in Minkowski space the metric tensor has signature $(3,-1,0)$, and the matrix $G$ from the first equation there has the following form:
$$
G = \begin{pmatrix}-1 & 0 & 0 & 0\\  0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{pmatrix}.
$$
Thus for two vectors $\vec{s}_1  = (t_1, x_1, y_1, z_1)^T$ and $\vec{s}_2  = (t_2, x_2, y_2, z_2)^T$ the scalar product has the form
$$
\vec{s}_1\cdot\vec{s}_2 = -t_1t_2 + x_1x_2 + y_1y_2 + z_1z_2.
$$
