Two-dimensional cross product ($u_x{v}_y-u_y{v}_x$)

For planar vectors $u=(u_x,u_y)$ and $v=(v_x,v_y)$, the scalar $u\times v=u_x{v}_y-u_y{v}_x$ is the determinant of their coordinate columns. It equals the perpendicular component of the three-dimensional cross product after embedding the plane in three dimensions. Its sign records orientation, and its absolute value is twice the area of the triangle spanned by $u,v$.

For planar points $A,B,P$, the sign of $(B-A)\times (P-A)$, using the two-dimensional cross product, determines which side of the directed line from $A$ to $B$ contains $P$. Zero means collinearity. A point is strictly inside a nondegenerate triangle exactly when these signs for its three cyclically ordered sides are all strictly positive or all strictly negative.