Kev Sib Npaug Cov Khoom Siv Siv Cov Cheebtsam Vector Hauv Chav
Peb tuaj yeem xam cov khoom sib tshuam ncaj qha yog tias peb paub cov khoom ntawm cov vectors. Cov txheej txheem zoo ib yam li rau cov khoom dot . Ua ntej, peb muab cov vectors unit i , j , thiab k sib npaug . Cov khoom vector ntawm tib lub vectors unit yog xoom.
i x i = j x j = k x k = 0
Los ntawm kev xa mus rau qhov sib npaug ntawm vector multiplication uas tau muab ua ntej (A x B = AB kev txhaum θ) thiab cov khoom anti-commutative ntawm vector multiplication (A x B =– B x A), ces peb tau txais:
i x j = -j i = k
j x k = -k x j = i
k x i = – i x k = j
Tam sim no peb qhia cov vectors A thiab B raws li lawv cov khoom, rhuav tshem lawv cov khoom thiab siv cov khoom ntawm cov vectors unit.
A x IB = (Axi + Ayj + Azk) x(Bxi + Byj + Bzk)
A x B = Axi x Bxi + Axi x Byj + Axi x Bzk +
Ayj x Bxi + Ayj x Byj + Ayj x Bzk +
Azk x Bxi + Azk x Byj + Azk x Bzk
A x B = AxBx (i x i) + AxBy (i x j) + Ax Bz (i x k) +
AyBx (j x i) + AyBy (j x j) + AyBz (j x k) +
AzBx (k x i) + AzBy (k x j) + AzBz (k x k)
Vim i x i = j x j = k x k = 0 dan i x j =–j x i = k, j x k =–k x j = i, k x i = -i x k = j, yog li ntawd:
A x B = AxBx (0) + AxBy (k+ Ax Bz (-j+
AyBx (-k+ AyBy (0) + AyBz (i+
AzBx (j+ AzBy (-i+ AzBz (0)
A x B = AxBy (k+ Ax Bz (-j+
A y B x ( -k ) + A y B z ( i ) +
A z B x ( j ) + A z B y ( -i )
A x B = AxBy (k+ Ax Bz (-j+ AyBx (-k+ AyBz (i+ AzBx (j+ AzBy (-i)
A x B = (AyBz - AzBy)i + (AzBx - Ax Bz)j + (AxBy - AyBx )k
Yog tias C = A x B ces cov khoom ntawm C yog raws li nram no:
Cx = A y B z – A z B y
Cy = A z B x – A x B z
Cz = A x B y – A y B x
