关于矢量点积-叉积分配律的证明-付云浩

ChinaEducationInnovationHerald2011NO.25ChinaEducationInnovationHerald,()ABCABAC⋅+=⋅+⋅vvvvvvv()ABCABAC×+=×+×vvvvvvv,,,Xv,Xijk=α+β+γvvvv()()()()()()()AAABBBCCCABCABCABCABCijkijkijk⎡⎤⋅+=α+β+γ⋅α+β+γ+α+β+γ⎣⎦=αα+α+ββ+β+γγ+γvvvvvvvvvvvv,,/,,/,AAAe=vv,AAv,AevAv;AABBBeBe⊥=+vvv,BBe⊥vBvAv;,AABAB⋅=⋅vv(1)AAAABeAB⋅=⋅vv,0BABe⊥⋅=vv(2)(1)(2),()AABAAABABABeBeABABeABe⊥⊥⋅=⋅+=⋅=⋅+⋅vvvvvvvvv(3)(3),()ABCABAC⋅+=⋅+⋅vvvvvvv,,,,AAAe=vv,AABBBeBe⊥=+vvv,,ABABABe⊥⊥×=⋅vvv(4),ABe⊥vAB×vv0AAABe×=vv,BABABeABe⊥⊥⊥×=⋅vvv(5)(4)(5),()AABABAABABABeBeABeABeABe⊥⊥⊥⊥×=×+=⋅=×+×vvvvvvvvvv(6)(6),,,AACCCeCe⊥=+vvv,CevAv,Bev,()()()()()AABAACAAAABCABCABeBeCeCeABeCeBeCe⊥⊥⊥⊥×+=×+++⎡⎤⎣⎦=×+++⎡⎤⎣⎦vvvvvvvvvvvvv,AAAABeCe+vvBCBeCe⊥⊥+vvAv,(6)()()BCABCABeCe⊥⊥×+=×+vvvvvv(7),(6),AABABABeABe⊥×=×+×vvvvvvAACACACeACe⊥×=×+×vvvvvv,AABAACBCABACABeABeACeACeABeACe⊥⊥⊥⊥×+×=×+×+×+×=×+×vvvvvvvvvvvvvvvv(8)/:,,,/,,/,,/,/,,::G642:A:1673-9795(2011)09(a)-0097-01,(7)(8),()ABCABAC×+=×+×vvvvvvv,()BCBCABeCeABeACe⊥⊥⊥⊥×+=×+×vvvvvvv(9)1,,AvBBe⊥vCCe⊥v,,BCBeCe⊥⊥+=⊕vvv,BCABeACe⊥⊥×+×=⊗vvvvvBBBeABe⊥⊥⊥×vvv,CCCeACe⊥⊥⊥×vvv,,⊕⊥⊗vv(),,()BCABeCe⊥⊥×+vvv//BCABeACe⊥⊥×+×vvvv(1,,BCBeCe⊥⊥+vvBCABeACe⊥⊥×+×vvvv222cosBCBeCeBCBC⊥⊥⊥⊥⊥⊥+=++θvv2cosBCABeACeABACABAC⊥⊥⊥⊥⊥⊥×+×=++⋅θvvvv()()22222cos2cosBCABeACeABACABACABCBC⊥⊥⊥⊥⊥⊥⊥⊥⊥⊥×+×=++⋅θ=++θvv,()BCABeCe⊥⊥×+vvvBCABeACe⊥⊥×+×vvvvBCABeACe⊥⊥×+×vvvv,(9),,()ABCABAC×+=×+×vvvvvvv/,,,,/,,,,,[1].[M].,2004.