Rauemi Whakarea Whakawhiti mā te whakamahi i ngā Wāhanga Waehere Ine
Ka taea e tātou te tatau tika i te hua whakawhiti mēnā e mōhio ana tātou ki ngā wāhanga o ngā whārite. He rite tonu te tukanga ki te hua ira . Tuatahi, ka whakareatia e tātou ngā whārite wae i , j , me k . Ko te hua whārite i waenga i ngā whārite wae kotahi he kore.
i x i = j x j = k x k = 0
Mā te kōrero ki te whārite whakarea whārite i ahu mai i mua (A x B = AB hara θ) me te āhuatanga ārai-whakarerekētanga o te whakareatanga whārite (A x B = – B x A), kātahi ka whiwhi tātou:
i x j = -j x i = k
j x k = -k x j = i
k x i = – i x k = j
Inaianei ka whakaatuhia e mātou ngā whārite A me B e ai ki ō rāua wāhanga, ka wetewetehia te hua o tā rāua hua, ā, ka whakamahia te hua o ngā whārite kotahi.
A x B = (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)
Na te mea 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, nā reira:
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
Mena ko C = A x B , ko ngā wāhanga o C penei:
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
