Ka Hoʻemi Vector: Nā Kumu, Nā Kānāwai, a me nā Noi
He manaʻo nui ka unuhi vector i ka makemakika, ke kino, a me ka ʻenekinia. I ke ola o kēlā me kēia lā, hālāwai pinepine mākou i nā kūlana kahi e pono ai mākou e unuhi i ʻelua a ʻoi aku paha nā vectors, no ka laʻana i ka helu ʻana i ke kuhikuhi o ka makani a i ʻole ka neʻe ʻana o nā mea. E kūkākūkā kēia ʻatikala i ka hōʻemi vector i ka hohonu, me kona wehewehe ʻana, nā loina kumu, nā kānāwai, a me nā noi ma nā ʻano like ʻole.
Wehewehena Vector
ʻO ka vector kahi nui i loaʻa ka nui (a i ʻole ka lōʻihi) a me ke kuhikuhi. ʻO nā hiʻohiʻona o nā vectors e komo pū me ka wikiwiki, ka wikiwiki, ka ikaika, a me ke kahua uila. Hōʻike pinepine ʻia nā vectors ma ke ʻano he mau pua ma nā kiʻikuhi, kahi e hōʻike ai ka lōʻihi o ka pua i ka nui a hōʻike ke kuhikuhi o ka pua i ke kuhikuhi o ka nui.
Ma ke ʻano makemakika, ua kākau pinepine ʻia nā vectors i nā ana ʻelua ma ke ʻano \( \mathbf{a} = (a_1, a_2) \) a i ʻole ma ke ʻano laulā \( \mathbf{a} = ai + bj \), kahi ʻo \( i \) a me \( j \) he mau vectors unit ma nā kuhikuhi x- a me y.
Ka Hoʻemi Vector: Nā Manaʻo Kumu
ʻO ka hoʻemi vector ke ʻano o ka hana o ka hoʻohui ʻana i nā vectors maikaʻi ʻole. Inā loaʻa iā mākou ʻelua mau vectors \( \mathbf{a} \) a me \( \mathbf{b} \), a laila ua like ka hoʻemi \( \mathbf{a} – \mathbf{b} \) me \( \mathbf{a} + (-\mathbf{b}) \). ʻO ka vector maikaʻi ʻole o ka vector \( \mathbf{b} \) he vector i loaʻa ka nui like akā ke kuhikuhi ʻē aʻe.
Ma ke ʻano makemakika, inā \( \mathbf{a} = (a_1, a_2) \) a me \( \mathbf{b} = (b_1, b_2) \), a laila:
\[ \mathbf{a} – \mathbf{b} = (a_1, a_2) – (b_1, b_2) = (a_1 – b_1, a_2 – b_2) \]
Laʻana o ka Vector Subtraction ma nā Ana ʻElua
Manaʻo mākou he ʻelua mau vectors i ʻelua mau ana, \( \mathbf{a} = (4, 3) \) a me \( \mathbf{b} = (1, 2) \). ʻO ka unuhi ʻana o kēia mau vectors ʻelua penei:
\[ \mathbf{a} – \mathbf{b} = (4 – 1, 3 – 2) = (3, 1) \]
Ka Hoʻemi Vector ma nā Ana ʻEkolu
Ua like ke kumumanaʻo o ka unuhi vector ma nā ana ʻekolu me ko nā ana ʻelua. Inā \( \mathbf{a} = (a_1, a_2, a_3) \) a me \( \mathbf{b} = (b_1, b_2, b_3) \), a laila:
\[ \mathbf{a} – \mathbf{b} = (a_1, a_2, a_3) – (b_1, b_2, b_3) = (a_1 – b_1, a_2 – b_2, a_3 – b_3) \]
Eia kekahi laʻana, inā ʻo \( \mathbf{a} = (5, 7, 2) \) a ʻo \( \mathbf{b} = (2, 3, 4) \), a laila ʻo ka unuhi ʻana penei:
\[ \mathbf{a} – \mathbf{b} = (5 – 2, 7 – 3, 2 – 4) = (3, 4, -2) \]
Ke Kānāwai o ka Hoʻemi Vector
Pili kekahi mau kānāwai kumu i ka unuhi vector, e like me nā kānāwai no ka hoʻohui vector. Eia nā kānāwai nui:
1. Commutative: ʻAʻole commutative ka hoʻemi vector, ʻo ia hoʻi:
\[ \mathbf{a} – \mathbf{b} \neq \mathbf{b} – \mathbf{a} \]
Eia kekahi laʻana, inā \( \mathbf{a} = (4,3) \) a me \( \mathbf{b} = (1,2) \):
\[ \mathbf{a} – \mathbf{b} = (4-1, 3-2) = (3,1) \]
ʻOiai:
\[ \mathbf{b} – \mathbf{a} = (1-4, 2-3) = (-3,-1) \]
2. Hoʻohui: ʻO ka hoʻemi vector i hui pū ʻia me ka hoʻohui he mea hoʻohui ia, ʻo ia hoʻi:
\[ \mathbf{a} – (\mathbf{b} – \mathbf{c}) = (\mathbf{a} – \mathbf{b}) + \mathbf{c} \]
Nā noi hoʻemi vector
Hoʻohana nui ʻia ka unuhi vector ma nā ʻano ʻepekema like ʻole a me nā ʻenekinia. Eia kekahi mau laʻana:
1. Ke Kino
I loko o ke kinoea, hoʻohana ʻia ka unuhi vector e hoʻoholo ai i ka ikaika hopena, ka manawa, ka neʻe ʻana, ka wikiwiki pili, a me nā mea hou aku. No ka laʻana, inā hana ʻelua mau ikaika ma kahi mea, hiki ke helu ʻia ka ikaika net me ka hoʻohana ʻana i ka unuhi vector. Manaʻo ʻia ʻelua mau ikaika \( \mathbf{F_1} \) a me \( \mathbf{F_2} \) e hana ma kahi mea ma nā ʻaoʻao ʻē aʻe; ua helu ʻia ka ikaika net \( \mathbf{F} \) penei:
\[ \mathbf{F} = \mathbf{F_1} – \mathbf{F_2} \]
2. ʻEnekinia a me ka ʻenehana
I loko o ka ʻenekinia kīwila, hiki ke hoʻohana ʻia ka unuhi vector e kālailai i nā ikaika e hana ana ma nā hale, e like me nā alahaka a i ʻole nā hale. No ka laʻana, hiki i nā ʻenekinia ke hoʻohana i ka unuhi vector e hoʻoholo ai i ka ikaika e hana ana ma kahi kikoʻī i loko o kahi hale ma muli o kahi ukana i kau ʻia.
3. Hoʻokele a me ka Lewa
I ka hoʻokele ʻana i ka lewa a me ke kai, he mea nui ka unuhi vector no ka hoʻokele ʻana i nā ala mai kekahi wahi a i kekahi, ʻoiai inā he mau haunaele makani a i ʻole nā au o ke kai. No ka laʻana, inā e lele ana kahi mokulele i kahi wikiwiki e kū pono ana i ka makani, hoʻohana ʻia ka unuhi vector e hoʻoholo ai i ka wikiwiki maoli a me ke poʻo o ka mokulele.
4. Nā ʻōnaehana Robotics a me nā ʻōnaehana hoʻokele
I loko o ka robotics, hoʻohana ʻia ka vector subtraction no ka hoʻolālā ala a me ka pale ʻana i nā mea keakea. Pono nā robots e helu pololei i ko lākou kūlana e pili ana i ko lākou kaiapuni.
Nā Laʻana o ka Hoʻopili ʻana o ka Vector Subtraction
Manaʻo ʻia e neʻe ana kahi moku me ka wikiwiki \( \mathbf{v_ship} \) a kuhikuhi ʻia e kūʻē i ke au o ka wai e neʻe ana me ka wikiwiki \( \mathbf{v_current} \). No ka hoʻoholo ʻana i ka wikiwiki holoʻokoʻa o ka moku e pili ana i ka honua, hiki iā mākou ke hoʻohana i ka vector subtraction:
\[ \mathbf{v_total} = \mathbf{v_ships} – \mathbf{v_current} \]
Manaʻo ʻia \( \mathbf{v_kapal} = (10, 15) \) km/h a me \( \mathbf{v_arus} = (2, 3) \) km/h, a laila:
\[ \mathbf{v_total} = (10 – 2, 15 – 3) = (8, 12) \] km/h.
Ka hopena
He hana koʻikoʻi ka hoʻemi ʻana o ka vector me nā noi koʻikoʻi ma nā ʻano kahua like ʻole. ʻO ka hoʻomaopopo maikaʻi ʻana i kāna mau loina kumu a me nā noi e hiki ai iā mākou ke hoʻoponopono i nā pilikia paʻakikī i ka physics, ka ʻenekinia, a me nā kahua ʻē aʻe. Ma ka hoʻomaopopo ʻana i nā manaʻo kumu, nā kānāwai, a me nā noi o ka hoʻemi ʻana o ka vector, hiki iā mākou ke hana maʻalahi i ka nānā ʻana a me nā helu ʻana e pono ai ma nā ʻano kūlana ʻoihana a me ka ʻepekema.
