Dot Product for 2D Vectors

We already know how to add and subtract vectors—placing them tip-to-tail or using components. But what about multiplying two vectors? Simply multiplying their magnitudes $|\vec{A}| \cdot |\vec{B}|$ loses all directional information. We need a multiplication that captures how vectors relate to each other geometrically.

The dot product (or scalar product) provides exactly this. It combines two vectors $\vec{A}$ and $\vec{B}$ to produce a scalar that measures their alignment:

$$\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos\theta$$

Here $|\vec{A}|$ and $|\vec{B}|$ are the magnitudes of the vectors, and $\theta$ is the angle between them when placed tail-to-tail.