1. ## Law of Cosines

If $\bold{C} = \bold{A} + \bold{B}$

Then $\bold{C} \cdot \bold{C} = (\bold{A} + \bold{B}) \cdot (\bold{A} + \bold{B})$

$|\bold{C}|^{2} = |\bold{A}^{2}| + |\bold{B}^{2}| + 2|\bold{A}||\bold{B}| \cos \theta$ where $\theta$ is the angle between the extension of $\bold{A}$ and $\bold{B}$.

Then this is usually expressed as $C^{2} = A^{2} + B^{2} - 2AB \cos \phi$ where $\phi$ is the angle between $\bold{A}$ and $\bold{B}$.

But why isn't $\bold{C} \cdot \bold{C} = |\bold{C}|$?

2. Because say $\bold{C} = (x,y)$ then $|\bold{C}| = \sqrt{x^2+y^2}$ and $\bold{C}\cdot \bold{C} = x^2+y^2$. So you are off by a square root.