dot product

The dot product takes two vectors, multiplies their matching coordinates, and adds the results into a single number: (a,b) . (c,d) = a*c + b*d. So (1,2) . (3,4) = 1*3 + 2*4 = 11.

That one number is a measure of how much the two vectors point the same way. Positive means they roughly agree in direction, zero means they are perpendicular (no shared direction at all), and negative means they oppose each other.

It also links length and angle: u . v = |u| * |v| * cos(theta), where theta is the angle between them. So once you know the dot product and the two lengths, you can recover the angle.