Orientation (Vectors)

Let $\theta$ be the Angle between two Vectors. If $0<\theta <\pi$, the Vectors are positively oriented. If $\pi < \theta < 2\pi$, the vectors are negatively oriented.

Two vectors in the plane

$$\left[{\matrix{x_1\cr x_2\cr}}\right] \hbox{and } \left[{\matrix{y_1\cr y_2\cr}}\right]$$

are positively oriented Iff the Determinant

$$D\equiv \left\vert\matrix{x_1 & y_1\cr x_2 & y_2\cr}\right\vert > 0,$$

and are negatively oriented Iff the Determinant $D< 0$.