Mea Hoʻonui Kiko me ka hoʻohana ʻana i nā ʻĀpana Vector Unit
Hiki iā mākou ke helu pololei i ka huahana scalar inā ʻike mākou i nā ʻāpana x, y a me z o nā vectors A a me B (nā vectors i ʻike ʻia).
No ka hana ʻana i kahi huahana kiko ma kēia ʻano, hana mua mākou i kahi huahana kiko o nā vectors unit, ma hope o kēlā hōʻike mākou i nā vectors A a me B i kā lākou mau ʻāpana, hoʻokaʻawale i ka huahana a hoʻohana i ka huahana o nā vectors unit.
ʻO nā vectors unit i , j, a me k he kū pololei kekahi i kekahi, e maʻalahi ai ka helu ʻana. Ma ka hoʻohana ʻana i ka hoʻohālikelike hoʻonui scalar i loaʻa ma luna ( AB = AB cos theta ), loaʻa iā mākou:
i. i = j. j = k. k = (1)(1) cos 0 = 1
i. j = i. k = j. k = (1)(1) cos 90 o = 0
I kēia manawa, hōʻike mākou i nā vectors A a me B ma ke ʻano o kā lāua mau ʻāpana, hoʻokaʻawale i kā lāua huahana a hoʻohana i ka huahana o nā vectors unit.
A . B = Axi . Bxi + Axi . Byj + Axi . Bzk +
Ayj . Bxi + Ayj . Byj + Ayj . Bzk +
Azk . Bxi + Azk . Byj + Azk . Bzk
A . B = AxBx (i . i) + AxBy (i . j) + Ax Bz (i . k) +
AyBx (j . i) + AyBy (j . j) + AyBz (j . k) +
AzBx (k . i) + AzBy (k . j) + AzBz (k . k)
Kārena i . i = j . j = k . k = 1 a me i . j = i . k = j . k = 0, a laila:
A . B = AxBx (1) + AxBy (0) + Ax Bz (0) +
AyBx (0) + AyBy (1) + AyBz (0) +
AzBx (0) + AzBy (0) + AzBz (1)
A. B = A x B x ( 1) + 0 + 0 +
0 + A y B y (1) + 0 +
0 + 0 + A z B z (1)
A . B = AxBx + AyBy + AzBz
Ma muli o nā hopena o kēia helu ʻana, hiki ke hoʻoholo ʻia ʻo ka huahana scalar a i ʻole ka huahana kiko o nā vectors ʻelua ka huina o nā huahana o kā lāua mau ʻāpana like.
Laʻana Nīnau 1:
Vector nui A a hiki B ʻo 5 a me 4 kēlā me kēia, e like me ka mea i hōʻike ʻia ma ke kiʻi ma lalo nei. ʻO 90 ke kihi i hoʻokumu ʻiaoE helu iā ia huahana kiko nā vectors ʻelua.
Pahana
Ma mua o ko kākou helu ʻana i ka huahana kiko o nā vectors A a me B, pono mua kākou e ʻike i nā ʻāpana o ka vector ʻelua.
A x = (5) cos 0 o = (5) (1) = 5
A y = (5) sin 0 o = (5) (0) = 0
ʻAz = 0
B x = (4) cos 90 o = (4) (0) = 0
B y = (4) hewa 90 o = (4) (1) = 4
Bz = 0
Loaʻa i ka Vector A nā ʻāpana vector ma ke axis-x wale nō a ʻo ka vector B nā ʻāpana vector ma ke axis-y wale nō. ʻO ka ʻāpana z he zero no ka mea aia nā vectors A a me B ma ka mokulele xy.
I kēia manawa, ke helu nei mākou i ka huahana kiko ma waena o nā vectors A a me B me ka hoʻohana ʻana i ka hoohalike huahana kiko me nā vectors ʻāpana:
A. B = Ax Bx + AyBy + AzBz
A. B = (5) (0) + (0) (4) + 0
A. B = 0 + 0 + 0
A. B = 0
E hoʻohālikelike kākou iā ia me ke ʻano hana mua
AB = AB cos theta
AB = (4)(5) cos 90
AB = (4) (5) (0)
AB = 0
Ua like ka hopena.
Laʻana Nīnau 2:
Vector nui A a hiki B he 5 a me 4 kēlā me kēia, e like me ka mea i hōʻike ʻia ma ke kiʻi ma lalo nei. E helu huahana kiko nā vectors ʻelua, inā he 30 ke kihi i hoʻokumu ʻiao
Pahana
Ma mua o ko kākou helu ʻana i ka huahana kiko o nā vectors A a me B, pono mua kākou e ʻike i nā ʻāpana o ka vector ʻelua.
ʻO ka ʻāpana z he ʻole no ka mea aia nā vectors A a me B ma ka mokulele xy.
I kēia manawa, ke helu nei mākou i ka huahana kiko ma waena o nā vectors A a me B me ka hoʻohana ʻana i ka hoohalike huahana kiko me nā vectors ʻāpana:
E hoʻohālikelike me ke ʻano hana mua.
