Nā Vectors a me nā ʻōnaehana hoʻonohonoho: Ke kahua o ka makemakika hou
Pendahuluan
I ka makemakika a me ka ʻepekema, ʻo nā manaʻo o nā vectors a me nā ʻōnaehana hoʻonohonoho he mau kumu koʻikoʻi e hiki ai ke hoʻomaopopo a hoʻoponopono i nā pilikia ma nā kahua e like me ke kino, ka ʻenekinia, a me ka ʻepekema kamepiula. E nānā hou kēia ʻatikala i nā manaʻo kumu o nā vectors a me nā ʻōnaehana hoʻonohonoho, a me kā lākou noi ʻana ma nā ʻano aʻo like ʻole.
Nā Vectors: Wehewehena a me ka Hoʻokaʻawale ʻana
I ka ʻōlelo maʻalahi, he mea makemakika ka vector nona ka nui a me ke kuhikuhi. Hoʻokaʻawale kēia iā ia mai kahi scalar, nona ka nui wale nō akā ʻaʻohe kuhikuhi. I ka makemakika, hōʻike pinepine ʻia nā vectors e nā pua ma kahi ākea ʻelua-dimensional (2D) a ʻekolu-dimensional (3D), kahi e hōʻike ai ka lōʻihi o ka pua i ka nui a hōʻike ke kuhikuhi o ka pua i ke kuhikuhi.
Nā ʻAno o nā Vectors
1. Vector Kūlana: He vector e hōʻike ana i kahi o kahi kiko ma ka lewa e pili ana i ke kumu.
2. Vector Velocity: Hōʻike i ka wikiwiki o ka loli ʻana o ke kūlana o kahi mea i ka manawa.
3. Vector Force: He vector e hōʻike ana i ka nui o ka ikaika a me ke kuhikuhi e hana ai ka ikaika ma luna o kahi mea.
4. Vector Unit: He vector me ka lōʻihi o hoʻokahi unit e hōʻike ana i kahi kuhikuhi ma ka lewa.
Ka Hoʻopaʻa ʻana i nā Vector a me nā Hana
Hōʻikeʻike
Ma kahi ākea ʻelua-dimensional, kākau pinepine ʻia nā vectors ma ke ʻano \( \mathbf{v} = (v_1, v_2) \), a ma kahi ākea ʻekolu-dimensional, kākau ʻia lākou e like me \( \mathbf{v} = (v_1, v_2, v_3) \). No ka laʻana, ʻo ka vector \( \mathbf{v} = (3, 4) \) he ʻāpana o 3 ma ka axis-x a he ʻāpana o 4 ma ka axis-y.
Hoʻohui a me ka Hoʻemi Vector
Hoʻohui ʻia nā vectors ʻelua ma ka hoʻohui ʻana i kā lākou mau ʻāpana. No ka laʻana, inā \( \mathbf{u} = (u_1, u_2) \) a me \( \mathbf{v} = (v_1, v_2) \), a laila \( \mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2) \). Hoʻohui ʻia ka unuhi ʻana ma ke ʻano like: \( \mathbf{u} – \mathbf{v} = (u_1 – v_1, u_2 – v_2) \).
Hoʻonui Scala
ʻO ka hoʻonui scalar e pili ana i ka hoʻonui ʻana i kahi vector me kahi helu maoli. Inā \( \mathbf{v} = (v_1, v_2) \) a he scalar ʻo k, a laila \( k\mathbf{v} = (kv_1, kv_2) \).
Huahana Kiko a me Huahana Keʻa
I loko o ka hakahaka ʻekolu-dimensional, aia ʻelua mau hana koʻikoʻi e pili ana i nā vectors ʻelua: ka huahana kiko a me ka huahana keʻa.
Huahana Kiko: \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \). ʻO ka hopena o ka huahana kiko he scalar a he ana ia o ka hana o kekahi vector ma ke kuhikuhi like me kekahi.
Huahana Keʻa: ʻO \( \mathbf{u} \times \mathbf{v} \) e hana i kahi vector hou i orthogonal (perpendicular) i nā vectors mua ʻelua. ʻOi aku ka paʻakikī o kāna hōʻike algebraic, akā he mea nui loa ia i ka physics, ʻoi aku hoʻi i ka hoʻoholo ʻana i ka torque a i ʻole ka moment o ka ikaika.
ʻŌnaehana Hoʻonohonoho: Manaʻo a me nā ʻAno
ʻO ka ʻōnaehana hoʻonohonoho he ʻōnaehana i hoʻohana ʻia e hoʻoholo ai i ke kūlana o kahi kiko ma ka lewa. Nui nā ʻano ʻōnaehana hoʻonohonoho like ʻole, akā ʻo ka mea maʻamau ka Cartesian, polar, a me nā ʻōnaehana hoʻonohonoho cylindrical.
ʻŌnaehana Hoʻonohonoho Cartesian
ʻO ka ʻōnaehana hoʻonohonoho Cartesian ka ʻōnaehana i hoʻohana nui ʻia, ʻoi aku hoʻi i ka makemakika kumu a me ke kino. Ma kēia ʻōnaehana, ua hoʻoholo ʻia ke kūlana o kēlā me kēia kiko ma ka lewa e kona mamao mai ʻelua a ʻekolu paha mau mokulele kuhikuhi kū pololei.
– 2D: Ma kahi ākea ʻelua-dimensional, ua hoʻoholo ʻia kēlā me kēia kiko \( (x, y) \) e kona mamao mai ka axis-x a me ka axis-y.
– 3D: Ma ka hakahaka ʻekolu-dimensional, hoʻohana ke kiko \( (x, y, z) \) i kahi z-axis hou aʻe e hoʻoholo ai i ke kūlana.
Nā ʻōnaehana hoʻonohonoho polar a me cylindrical
Nā Hoʻonohonoho Polar: Hoʻohana nui ʻia kēia ʻōnaehana i nā pilikia e pili ana i ka symmetry radial. Ma nā hoʻonohonoho polar, ua wehewehe ʻia kēlā me kēia kiko e kona mamao radial (r) mai ke kumu a me kahi kihi \( \theta \) i ana ʻia mai ka axis x maikaʻi.
\[ (r, \theta) \]
Nā Hoʻonohonoho Cylindrical: He hui pū ʻana o nā hoʻonohonoho Cartesian a me polar, e hoʻohana ana iā \((r, \theta) \) e kuhikuhi i ke kūlana ma kahi mokulele a me z no ke kiʻekiʻe. Hoʻohana pinepine ʻia i nā pilikia physics e pili ana i nā mea e hoʻohuli ana e like me ke kahe ʻana o ka wai i loko o nā paipu.
Nā noi Vector a me nā ʻōnaehana hoʻonohonoho
ʻO ke kinoea
He mea nui nā vectors i ka physics. ʻO ka wikiwiki, ka wikiwiki, a me ka ikaika he mau manaʻo kino i hōʻike ʻia e nā vectors. No ka laʻana, hiki ke hōʻike ʻia ke kānāwai ʻelua o Newton ma ke ʻano vector: \( \mathbf{F} = m\mathbf{a} \), kahi \( \mathbf{F} \) ka ikaika, \( m \) ka nuipa, a ʻo \( \mathbf{a} \) ka wikiwiki.
ʻEnekinia a me ka ʻenehana
Ma nā ʻano aʻo ʻenekinia like ʻole, hoʻohana ʻia ka nānā ʻana o ka vector e hoʻomaʻalahi i nā helu paʻakikī. No ka laʻana, ʻo ka nānā ʻana i ke ʻano hana i ka ʻenekinia kīwila e pili ana i ka hoʻohui ʻana i nā vectors ikaika e hana ana ma kahi ʻōnaehana e hoʻoholo ai i nā stress a me nā deformations.
ʻEpekema Kamepiula a me nā Kiʻi
I nā kiʻi kamepiula, hoʻohana ʻia nā ʻōnaehana hoʻonohonoho e wehewehe i ke kūlana o nā pikela ma ka pale. ʻO nā hoʻololi vector kekahi kumu o ka animation 3D, kahi e neʻe ai nā mea, hoʻohuli, a hoʻololi i ke ʻano ma o nā hana vector a me matrix.
Hoʻololi Hoʻonohonoho
ʻO ka hoʻololi hoʻonohonoho e pili ana i ka neʻe ʻana i kahi kiko mai kekahi ʻōnaehana hoʻonohonoho i kekahi. He mea pono kēia i nā kūlana he nui, e like me ka hoʻololi ʻana i ke kumu i ka algebra linear a i ʻole ka hoʻohuli ʻana i kahi mea i nā kiʻi 3D.
Ka hopena
He mea nui nā vectors a me nā ʻōnaehana hoʻonohonoho i ka makemakika a me nā ʻano aʻo ʻepekema like ʻole. ʻO ka hoʻomaopopo ʻana iā lākou e hiki ai ke hoʻoponopono i nā ʻano pilikia helu a me nā pilikia loiloi paʻakikī. Mai ka hoʻoholo ʻana i ke kūlana o nā mea i ka lewa a hiki i ka wehewehe ʻana i nā hanana kino, he mau mea hana pono lākou i ka arsenal makemakika hou. Me ke aʻo hohonu ʻana, e hoʻomau ka hoʻonui ʻana o nā noi o nā vectors a me nā ʻōnaehana hoʻonohonoho, e hoʻokuke ana i nā palena o ka ʻike kanaka.
