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【新运算】平面向量的复数积

Floor 1 巨大八爪鱼 12/18/14 23:12
复数的乘法公式为:(a+bi)(c+di)=(ac-bd)+(bc+ad)i
对于向量(a,b)和向量(c,d),定义与其类似的运算“⊕”,其运算规则如下:
(a, b) ⊕ (c, d) = (ac - bd, bc + ad)
并称之为平面向量的“复数积”
Floor 2 巨大八爪鱼 12/18/14 23:16
如果改用向量的极坐标式表示上述运算规则,那么有:
xvec A ⊕ yvec B = xyvec(A + B)

特别地,当y=1时,
xvec A ⊕ vec B = xvec(A + B)
可见,利用向量的复数积,可以不改变一个向量的模长,而任意改变该向量的方向。
Floor 4 巨大八爪鱼 12/18/14 23:19
例如:4vec 45° ⊕ vec 1° = 4vec 46°
Floor 5 巨大八爪鱼 12/20/14 21:25
回复:2楼
证明:
vec A ⊕ vec B = (cos A, sin A) ⊕ (cos B, sin B)
=(cosAcosB - sinAsinB, sinAcosB + cosAsinB)
=(cos(A+B), sin(A+B))
=vec(A+B)

xvec A ⊕ yvec B = xy vec A ⊕ vec B = xyvec(A+B)
Floor 6 巨大八爪鱼 12/20/14 21:26
tcom(vec 90°) = i
这就是为什么一个复数乘以i可以在复平面上旋转90°的原因
Floor 7 巨大八爪鱼 12/20/14 21:29
i² = -1 对应 vec 90°⊕vec 90° = vec 180°
i³ = -i 对应 vec 90°⊕vec 90°⊕vec 90° = vec 270°

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