What is the area of a 3D polygon p₁p₂...pₙp₁? If the polygon is non-planar, this is not well-defined: if you split up the polygon into triangles pᵢpⱼpₖ and add up their areas ½‖(pⱼ−pᵢ)×(pₖ×pᵢ)‖, the result depends on the choice of triangulation. However, if you forget to take norms and just add up the vectors ½(pⱼ−pᵢ)×(pₖ×pᵢ) instead, you always get the same result: the "vector area" of the polygon! So if you're dealing with non-planar polygons, or non-planar curves in general, it makes sense to think of area as a vector rather than a scalar.
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