Hoʻonui ʻana i nā kiko ma nā Vectors
ʻO ka huahana kiko (i kapa ʻia hoʻi he huahana kiko a i ʻole huahana scalar) kekahi o nā hana koʻikoʻi loa i ka makemakika vector. ʻIke pinepine ʻia kēia hana i ka physics, ka ʻenekinia, nā helu helu, ka ʻepekema kamepiula (e.g., ke aʻo ʻana i ka mīkini), a me nā kiʻi kamepiula. ʻAʻole e like me ka hoʻonui maʻamau, kahi e hana ai i kahi helu mai ʻelua mau helu, lawe ka huahana kiko i ʻelua mau vectors ma ke ʻano he mea hoʻokomo a hana i kahi scalar (kahi helu hoʻokahi). Ma o ka huahana kiko, hiki iā mākou ke hoʻomaopopo i ka pilina ma waena o ʻelua mau vectors: inā paha ma ke kuhikuhi like lākou, nā kuhikuhi ʻē aʻe, a i ʻole ke kū pololei kekahi i kekahi.
Ke Hoʻomaopopo nei i ka Hoʻonui ʻana o nā Kiko
Ma keʻano laulā, inā loaʻa iā mākou ʻelua vectors a a me b, ua kākau ʻia ka huahana kiko penei:
he · b
ʻO ka hopena, he helu scalar e hōʻike ana i ka "nui" o ke vector a ma ke kuhikuhi like me ka vector b. Hiki ke hoʻomaopopo ʻia kēia manaʻo e ke kiʻi aʻe: ke hoʻolālā mākou i ka vector a ma luna o ka vector b, ana mākou i ka nui o ka ʻāpana a "e holo ana" ma ke kuhikuhi o b. ʻO ka nui o nā vector ʻelua ma ke kuhikuhi like, ʻoi aku ka nui o kā lākou huahana kiko; ʻo ka ʻoi aku o ko lākou kūʻē ʻana, ʻoi aku ka liʻiliʻi (ʻoiai maikaʻi ʻole) o kā lākou huahana; a inā he perpendicular lākou, e lilo ka huahana kiko i zero.
ʻO ke ʻano hoʻonui kiko ma o nā ʻāpana
Manaʻo ʻia he vector ʻelua-dimensional:
a = (a₁, a₂)
b = (b₁, b₂)
No laila, ʻo ka hoʻonui ʻana o ke kiko:
a · b = a₁b₁ + a₂b₂
No nā vectors ʻekolu-dimensional:
a = (a₁, a₂, a₃)
b = (b₁, b₂, b₃)
No laila:
a · b = a₁b₁ + a₂b₂ + a₃b₃
ʻOi aku ka laulā, no kahi vector n-dimensional:
a · b = Σ (aᵢ bᵢ) no i = 1 a i n.
He maʻalahi kēia ʻano hana akā ikaika. Hoʻonui wale mākou i nā huaʻōlelo like a laila hoʻohui pū iā lākou. ʻO ia ke kumu i hoʻohana nui ʻia ai ka hoʻonui kiko i ka helu ʻana: he maʻalahi ke hoʻokō a he pono hoʻi.
ʻO ke ʻano hoʻonui kiko me ke kihi ma waena o nā vectors
Ma waho aʻe o nā ʻāpana, hiki ke hōʻike ʻia ka huahana kiko ma ke ʻano angular (geometric). Inā ʻo θ ke kihi ma waena o nā vectors a a me b , a laila:
a · b = |a| |b| cos(θ)
ʻIkepili:
– ʻO |a| ka lōʻihi (ka nui) o ka vector a
– ʻo |b| ka lōʻihi o ka vector b
– ʻo cos(θ) ke cosine o ke kihi ma waena o nā vectors ʻelua
Hoʻokūpaʻa kēia haʻilula i ke ʻano geometric: ana ka huahana kiko i ka "hoʻonohonoho" ʻana o nā kuhikuhi o ʻelua vectors. Ke liʻiliʻi ʻo θ (kokoke i ka 0°), kokoke ʻo cos(θ) i ka 1, no laila he nui a maikaʻi ka huahana kiko. Ke θ = 90°, cos(90°)=0, no laila he ʻole ka huahana kiko. Ke > 90°, he maikaʻi ʻole ʻo cos(θ), no laila he maikaʻi ʻole ka huahana kiko.
Laʻana Helu Huahana Kiko
Laʻana 1 (2D Vector)
ʻo kahi laʻana:
– a = (2, 3)
– b = (4, -1)
No laila:
a b = 2·4 + 3·(-1) = 8 – 3 = 5
ʻO ka hopena he 5 (scalar). ʻO ke ʻano kēia, ma ke ʻano holoʻokoʻa, aia nō kahi ʻāpana o ke kuhikuhi o ka vector a ma ke kuhikuhi like me b, ʻoiai kekahi o nā ʻāpana e kūʻē ana.
Laʻana 2 (Kūlike)
ʻo kahi laʻana:
– a = (1, 2)
– b = (2, -1)
Kiko:
a b = 1·2 + 2·(-1) = 2 – 2 = 0
No ka mea ʻo ka huahana kiko he 0, ua kū pololei (orthogonal) nā vectors ʻelua.
Nā Waiwai o ka Hoʻonui ʻana o nā Kiko
He nui nā ʻano koʻikoʻi o ka hoʻonui ʻana o nā kiko:
1. Hoʻololi
a · b = b · a
2. Hoʻokaʻawale i ka hoʻohui
a · (b + c) = a · b + a · c
3. Hoʻohui me ka hoʻonui scalar
(ka) · b = k (a · b) no ke k scalar
4. Hua kiko me ka vector zero
he · 0 = 0
5. Pilina me ka lōʻihi vector
he · he = |a|²
He mea pono loa kēia no ka mea mai ʻaneʻi hiki iā mākou ke loaʻa ka lōʻihi o ka vector:
|a| = √(a · a)
ʻO kēia mau waiwai e hoʻolilo i ka huahana kiko i hana kumu i ka algebra linear e kūkulu ʻia ma luna o nā manaʻo holomua he nui.
Ke ʻano a me ka wehewehe ʻana o ka Geometric
Hiki ke unuhi ʻia ka huahana kiko ma ke ʻano he ana o ka projection. Inā makemake mākou e ʻike i ka projection o ka vector a ma ke kuhikuhi o b, hiki iā mākou ke hoʻohana i ka huahana kiko. Hiki ke loaʻa ka ʻāpana o ka vector a ma ke kuhikuhi like me b ma o:
Ka hoʻolālā scalar o a a i b:
comp_b(a) = (a · b) / |b|
Ka hoʻolālā ʻana o ka vector a a i b:
proj_b(a) = ((a · b) / |b|²) b
He mea pono loa kēia wehewehe ʻana, no ka laʻana i ka helu ʻana i ke aka o kahi ikaika ma kekahi kuhikuhi a i ʻole i ka wā e wāwahi ai i ka neʻe ʻana i nā ʻāpana moe a me ke kū.
Nā Hoʻohana o ka Hoʻonui ʻana o nā Dot i ke Ola a me ka ʻEpekema
1. ʻIke Kino (Hana a me ka Ikehu)
I loko o ka physics, ua wehewehe ʻia ka hana e ka ikaika penei:
W = F · s = |F||s|cos(θ)
kahi ʻo F ka ikaika a ʻo s ka neʻe ʻana. Inā like ka ikaika me ka neʻe ʻana, he maikaʻi ka hana; inā aia ma ke kuhikuhi ʻē aʻe, he maikaʻi ʻole ka hana; inā kū pololei ia (e.g., ka ikaika maʻamau ma kahi mokulele), ʻaʻohe ka hana.
2. Ke hoʻoholo nei i ke kihi ma waena o nā vectors
Inā ʻike kākou iā a · b, |a|, a me |b|, a laila hiki ke helu ʻia ke kihi θ:
cos(θ) = (a · b) / (|a||b|)
A laila hiki ke loaʻa iā θ me ka hana inverse cosine (arccos).
3. Ke aʻo ʻana i ka mīkini a me ka ʻepekema ʻikepili
Hoʻohana ʻia ka huahana kiko e helu i ka helu a i ʻole ke ʻano like ma waena o ʻelua mau vector hiʻona. I loko o nā kumu hoʻohālike linear, lawe pinepine ka wānana i ke ʻano:
y = w · x + b
kahi ʻo w ke kaumaha a ʻo x ka vector hoʻokomo. ʻO ka huahana kiko ma aneʻi ke kikowaena o ka ʻōnaehana hoʻoholo.
4. Nā Kiʻi Kamepiula (Kukui)
I ka hōʻike ʻana i ka 3D, hoʻohana ʻia ka huahana kiko e hoʻoholo ai i ka ikaika o ke kukui ma luna o kahi ʻili. Pili pinepine ka ikaika i ka cosine o ke kihi ma waena o ke kuhikuhi o ke kukui a me ka maʻamau o ka ʻili. Me ka huahana kiko, lilo ka helu ʻana:
ʻO wau ∝ n · l
me n ka ʻili maʻamau a me l ke kuhikuhi o ka mālamalama (i hoʻomaʻamaʻa pinepine ʻia).
Nā hewa maʻamau e pale aku ai
ʻO kekahi mau hewa maʻamau i ke aʻo ʻana i nā huahana kiko:
– Ke manaʻo nei he vector ka hopena o ka huahana kiko (ʻoiai he scalar maoli ia).
– Ka hoʻohui ʻana o nā ʻāpana hewa (pono nā ʻāpana kūpono).
– Ua poina iā ia he hiki i ka huahana kiko ke lilo i mea maikaʻi ʻole.
– Ke hoʻohana nei i nā haʻilula kihi me ka ʻole o ka hōʻoia ʻana ua pololei ka vector i hoʻohana ʻia a me ka nui i helu ʻia.
– Ke kuhi hewa ʻana he ʻano kū pololei ka hua hoʻonui kiko ʻole (he ʻoiaʻiʻo kēia), akā inā ʻaʻole nā vectors ʻelua he mau vectors ʻole.
Pani
ʻO ka huahana kiko o nā vectors kahi manaʻo kumu e hoʻopili ai i ka algebra a me ke geometry. Me ka huahana kiko, ʻaʻole hiki iā mākou ke helu wale i ka helu o ʻelua vectors, akā e hoʻomaopopo pū i ka pilina kuhikuhi ma waena o lākou: inā paha lākou ma ke kuhikuhi like, nā kuhikuhi ʻē aʻe, a i ʻole orthogonal. ʻO kāna ʻano maʻalahi e maʻalahi ai ka hoʻopili ʻana i nā helu lima a me nā helu nui. Ma muli o kāna mau noi he nui ma nā ʻano like ʻole - mai ka physics a me ka nānā ʻikepili a hiki i nā kiʻi kamepiula - ʻo ka hoʻomaopopo ʻana i ka huahana kiko he ʻanuʻu koʻikoʻi no kekahi e aʻo ana i ka makemakika vector a me ka algebra linear.
Inā makemake ʻoe, hiki iaʻu ke hoʻohui i nā kiʻi, nā nīnau hoʻomaʻamaʻa me nā wehewehe, a i ʻole kahi mana o ka ʻatikala e kālele nui ana i kahi noi kikoʻī (physics, machine learning, a i ʻole geometry).
