Dot Product

$\vec{A}\cdot\vec{B} = \|\vec{A}\|\times\|\vec{B}\|\times\cos\theta$
$\vec{A}[x_{_1}, y_{_1}]\cdot\vec{B}[x_{_2}, y_{_2}] = x_{_1} \times x_{_2} + y_{_1} \times y_{_2}$

To calculate $\vec{A}\cdot\vec{B}$, pair up corresponding coordinates, multiply pairs, and add together.
$[1, 2]\cdot[3, 4] = 1·3+2·4 = 3+8 = 11$

To visualise $\vec{A}\cdot\vec{B}$, project one onto the other and multiply their lengths together i.e. project $\vec{A}$ onto $\vec{B}$ and multiply $\|\vec{A}_2\|\times\|\vec{B}\|$. If they point in a similar direction the value will be positive, if they are perpendicular to each other it will be zero, and if they are pointing away from each other it must be negative.

For normalised vectors the dot product can be seen as a measure of similarity because, for two normals, the magnitudes will both be $1$, meaning
$\widehat{\vec{A}}\cdot\widehat{\vec{B}}=\|\widehat{\vec{A}}\|\times\|\widehat{\vec{B}}\|\times\cos\theta$
will be
$\widehat{\vec{A}}\cdot\widehat{\vec{B}}=1\times1\times\cos\theta$
which means
$\widehat{\vec{A}}\cdot\widehat{\vec{B}}=\cos\theta$
and $\cos\theta=1$ at $\theta=0\degree$,
decreases to $\cos\theta=0$ at $\theta=90\degree$,
continues to $\cos\theta=-1$ at $\theta=180\degree$,
heads back to $\cos\theta=0$ at $\theta=270\degree$ ($-90\degree$),
and finally back up to $\cos\theta=1$ as $\theta$ approaches $360\degree$ which is also $0\degree$.