Dot product

A way to multiply two vectors that returns a number measuring how much they agree in direction.

The dot product $\mathbf{u}\cdot\mathbf{v}=u_1v_1+u_2v_2+u_3v_3$ equals $|\mathbf{u}||\mathbf{v}|\cos\theta$ for every pair of vectors — the same number computed two ways, one from components, one from magnitudes and the angle between them. The sign splits into three cases: positive means $\theta<90°$ , negative means $\theta>90°$ , and exactly zero means $\theta=90°$ — the two vectors are perpendicular.

Examples

For $\mathbf{u}=\langle 1,2,-2\rangle$ , $\mathbf{v}=\langle 2,-1,2\rangle$ : $\mathbf{u}\cdot\mathbf{v}=-4$ , giving $\theta\approx 116.4°$ .
A video game culls away-facing polygons by dotting a surface's normal vector with the direction to the camera.