82 CHAPTER 1. COORDINATES AND VECTORS −→ σ ( ABC ) = σ ( ABC ) −→ k : the oriented area is the vector −→ k times the signed area in our old sense. This interpretation can be applied as well to any oriented polygon contained in a plane in space. In particular, by analogy with Proposition 1.6.1 , we can defne a Function which assigns to a pair oF vectors −→ v , −→ w ∈ R 3 a new vector representing the oriented area oF the parallelogram with two oF its edges emanating From the origin along −→ v and −→ w , and oriented in the direction oF the frst vector. This is called the cross product 16 oF −→ v and −→ w , and is denoted −→ v × −→ w. ±or example, the sides emanating From A in △ ABC in ±igure 1.41 are represented by −→ v = −−→ AB = 2 −→ ı + −→ + −→ k −→ w = −→ AC = −→ ı + 2 −→ − −→ k ; these vectors, along with the direction oF
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