ʻO ke kūkākūkā ʻana i nā vectors i ka makemakika he mea hiki ʻole ke hoʻokaʻawale ʻia mai kahi kūkākūkā ʻana o ka ʻōnaehana hoʻonohonoho Cartesian. ʻO ka ʻōnaehana hoʻonohonoho Cartesian ka ʻōnaehana i hoʻohana pinepine ʻia no ke kumu hoʻohālike a me ka nānā ʻana i nā hanana like ʻole ma kahi ʻelua a me ʻekolu-dimensional. Ma kēia ʻatikala, e ʻimi mākou i ke kumumanaʻo o nā vectors like ma ke ʻano o ka ʻōnaehana hoʻonohonoho Cartesian.
Hoʻolauna i nā Vectors ma ka ʻōnaehana hoʻonohonoho Cartesian
I loko o kahi ʻōnaehana hoʻonohonoho Cartesian, hiki ke hōʻike ʻia kēlā me kēia kiko i kahi ākea ʻelua-dimensional ma ke ʻano he pālua i kauoha ʻia (x, y), kahi ʻo x ka hoʻonohonoho pae a ʻo y ka hoʻonohonoho kū. No kahi ākea ʻekolu-dimensional, loaʻa iā mākou ka triplet (x, y, z). ʻO kahi vector ma kēia pōʻaiapili he mea makemakika nona ka nui (a i ʻole ka lōʻihi) a me ke kuhikuhi.
ʻO ka maʻamau, hōʻike ʻia kahi vector ma kahi ākea ʻelua-dimensional e like me \(\vec{v}\) = (v_x, v_y), kahi ʻo \(v_x\) a me \(v_y\) nā ʻāpana o ka vector ma ke axis-x a me ke axis-y. Ma kahi ākea ʻekolu-dimensional, hōʻike ʻia kahi vector e like me \(\vec{v}\) = (v_x, v_y, v_z).
Ke Manaʻo o ke Kaulike Vector
Ua ʻōlelo ʻia ʻelua mau vectors he like inā like ko lākou nui a me ke kuhikuhi like, me ka nānā ʻole i ko lākou wahi hoʻomaka. Ma nā huaʻōlelo makemakika, ua ʻōlelo ʻia ʻelua mau vectors \(\vec{u}\) = (u_x, u_y) a me \(\vec{v}\) = (v_x, v_y) he like inā:
1. \(u_x = v_x\)
2. \(u_y = v_y\)
ʻO ke kumu, ʻaʻole i hoʻopaʻa ʻia nā vectors i kahi hoʻomaka kikoʻī. Hiki ke kau ʻia nā vectors ʻelua ma nā wahi āpau i ka lewa, akā inā like ko lākou kuhikuhi a me ka nui, ua manaʻo ʻia lākou he like, a like paha. He waiwai koʻikoʻi kēia e hoʻolilo ai i nā vectors i mea hana maʻalahi i ka makemakika a me ka physics.
Analogy Geometric
Manaʻo mākou he ʻelua vectors kā mākou \(\vec{u}\) = (3, 4) a me \(\vec{v}\) = (3, 4). ʻO kēia mau vectors ʻelua, ke nānā ʻia i loko o kahi ʻōnaehana hoʻonohonoho Cartesian, e hōʻike i nā pua e like ana ke kuhikuhi a like ka lōʻihi, ʻoiai e hoʻomaka paha lākou mai nā wahi like ʻole. No laila, inā e kahakiʻi mākou iā \(\vec{u}\) mai ke kumu (0, 0) a i ke kiko (3, 4) a me \(\vec{v}\) mai kahi kumu ʻē aʻe, e ʻōlelo (1, 1), a i ke kiko (4, 5), ua like mau kēia mau vectors ʻelua no ka mea ua like ke kuhikuhi a me ka nui o lākou.
Ka Hōʻike Vector like ma ka Makemakika
Ma ke ʻano makemakika, hahai nā vectors like i ke kumumanaʻo penei:
– Inā he vector ʻo \(\vec{v}\) = (v_x, v_y), a laila hiki ke loaʻa kekahi vector e like me \(\vec{v}\) ma ka hoʻohui ʻana i ka vector unuhi like i kona mau kiko hoʻomaka a me ka hopena.
– Ma ke ʻano maʻamau, inā ʻo \(\vec{v_1}\) = (v_{1x}, v_{1y}) a me \(\vec{v_2}\) = (v_{2x}, v_{2y}) he ʻelua mau vectors like, a laila aia kahi vector mau \(\vec{k}\) = (k_x, k_y) e like me:
\[
\vec{v_1} = \vec{v_2} + \vec{k} – \vec{k}
\]
Paʻa kēia lakio i ka hakahaka n-dimensional a hoʻokūpaʻa i ka ʻoiaʻiʻo e pili ana nā vectors i nā ʻokoʻa o nā kūlana, ʻaʻole nā kūlana ponoʻī.
Ka Hoʻohana ʻana o nā Vectors like i ka Physics
I loko o ke kinoea, he mea koʻikoʻi ke kumumanaʻo o nā vectors like, ʻoiai hoʻi i ka nānā ʻana i ka ikaika, ka wikiwiki, a me ka momentum. No ka laʻana, hiki ke unuhi ʻia nā ikaika e hana ana ma kahi kiko i hāʻawi ʻia i loko o kahi mea (e like me nā vectors like) inā hana lākou i ka hopena like ma ke ʻano o ka wikiwiki linear a i ʻole ka loli o ka momentum.
Nā Laʻana Noi:
1. Nā Mana like a me nā Vectors:
I loko o ka mechanics kuʻuna, inā hōʻike ʻia kahi ikaika F ma ke ʻano he vector a hoʻopili ʻia ma kekahi wahi ma kahi mea, hiki iā mākou ke hoʻoneʻe i ke kiko o ka hoʻopili ʻana o ka ikaika ma kahi mamao like. He mea nui kēia i ka helu ʻana i nā manawa o ka ikaika a i ʻole nā torques, kahi e hoʻohana ʻia ai nā ʻāpana ikaika like e hoʻoponopono i nā pilikia mechanical.
2. Māmā:
Hōʻike ka wikiwiki ma ke ʻano he vector i ke kuhikuhi a me ka wikiwiki o ka neʻe ʻana o kahi mea. No ka laʻana, hiki ke hōʻike ʻia ka wikiwiki o kahi kaʻa e neʻe ana i ka hikina ma 60 km/h ma ke ʻano he vector (60, 0) inā kuhikuhi ka axis-x i ka hikina. Hōʻike nā vectors like āpau i nā kūlana neʻe like, ʻoiai inā ʻokoʻa ko lākou mau wahi hoʻomaka, e.g., (1, 1) a i (61, 1).
Nā Hoʻonohonoho Homogeneous a me nā Vectors like
Ma kahi ākea ʻekolu-dimensional, hoʻohana pinepine mākou i nā hoʻonohonoho homogeneous e hoʻonui i kā mākou loiloi. Hoʻolauna kēia ʻōnaehana i ka matrix projection operator matrix, kahi e hoʻonui ai i ko mākou ʻike i nā vectors like. Hoʻohana pinepine ʻia nā hoʻonohonoho homogeneous i nā kiʻi kamepiula e hoʻomaʻalahi i nā hoʻololi geometric e like me nā rotations, nā unuhi, a me ka scaling. Ma kēia ʻano, ʻae nā vectors homogeneous iā mākou e hana i nā manipulations like a nani hoʻi ma nā hoʻonohonoho Cartesian.
Ka hopena
ʻO nā vectors like i loko o kahi ʻōnaehana hoʻonohonoho Cartesian kahi manaʻo nui e hoʻokumu i ke kumu o nā noi he nui o ka makemakika a me ka physics. ʻO ka hoʻomaopopo ʻana i kēia manaʻo e pili ana i ka ʻike ʻana he like nā vectors ʻelua inā like ko lākou kuhikuhi a me ka nui, ʻoiai he ʻokoʻa paha ko lākou mau wahi hoʻomaka. Hōʻike ka hōʻike makemakika o nā vectors like e hiki ai i kēia waiwai ke hoʻoneʻe i nā wahi hoʻomaka a me nā wahi hopena o nā vectors me ka ʻole o ka hoʻololi ʻana i ko lākou mau waiwai kumu.
ʻO ka hoʻohana ʻana o kēia manaʻo ma nā ʻano like ʻole, e like me ka physics, e hōʻike ana i ke koʻikoʻi o ka hoʻomaopopo ʻana i ke kumumanaʻo vector no ka nānā hou aku. I ke ao maoli, ʻo ke kumumanaʻo o nā vectors like e ʻae ai i nā helu maʻalahi o nā ikaika, nā wikiwiki, a me nā ʻano ʻē aʻe he nui o ka mechanics a me kinematics.
Me ka hiki ke hōʻike a hoʻoponopono i nā vectors i loko o kahi ʻōnaehana hoʻonohonoho Cartesian, hiki iā mākou ke hoʻohālike a kālailai i nā ʻano hanana paʻakikī a me nā ʻōnaehana me ke kiʻekiʻe o ka pololei a me ka pololei. ʻO kēia ka mea e hoʻolilo ai i ke kumumanaʻo o nā vectors like i kumuhana koʻikoʻi a hoihoi i ke aʻo ʻana i ka makemakika a me ka physics.
