已知向量OA=(a1,a2),OB=(b1,b2),设以向量OA,向量OB为邻边的平行四边形的面积为S,求证S^2=(a1b2-a2b1)^2

设OA OB夹角为x
我们有 OA·OB = |OA| * |OB| * cos x = a1b1 + a2b2
那么
S^2 = (|OA| * |OB| * sin x)^2
= |OA|^2 * |OB|^2 * (1 - cos^2 x)
= |OA|^2 * |OB|^2 - |OA|^2 * |OB|^2 * cos^2 x
= (a1^2 + a2^2) * (b1^2 + b2^2) - (a1b1 + a2b2)^2
= (a1b2 - a2b1)^2