Question

已知向量OA(a1,b1),OB(a2,b2),设以向量OA,OB为邻边的平行四边形的面积为S,求证S=|a1b2-a2

Answer 1

设OA OB夹角为θ

OA•OB = |OA|*|OB|*cosθ=a1a2+b1b2

S²= (|OA||OB|sinθ)²

=|OA|²|OB|²(1-cos²θ)

=|OA|²|OB|²-|OA|²|OB|²cos²θ

=(a1²+b1²)(a2²+b2²)-(a1a2+b1b2)²

=(a1b2-a2b1)²

所以S=|a1b2-a2b1|