Nā Vectors ʻElua-Dimensional i loko o kahi ʻōnaehana hoʻonohonoho
Pendahuluan
I ka makemakika a me ke kinoea, he manaʻo koʻikoʻi nā vectors a hoʻohana pinepine ʻia e hōʻike i nā nui me ka nui a me ke kuhikuhi. ʻO nā vectors ʻelua-dimensional, ʻo ia hoʻi, nā vectors i loko o ka mokulele, i hōʻike ʻia me ka hoʻohana ʻana i ʻelua mau ʻāpana hoʻonohonoho. E hāʻawi kēia ʻatikala i kahi ʻike hohonu o nā vectors ʻelua-dimensional i loko o kahi ʻōnaehana hoʻonohonoho, me kā lākou wehewehe ʻana, hōʻike, nā hana kumu, a me nā noi ma nā ʻano like ʻole.
Ka Wehewehena a me ka Hōʻikeʻike
Wehewehena Vector
ʻO ka vector kahi mea nona nā ʻano koʻikoʻi ʻelua: ka nui a me ke kuhikuhi. Ma kahi ʻōnaehana hoʻonohonoho ʻelua-dimensional (2D), hōʻike pinepine mākou i nā vectors ma ke ʻano he mau hui i kauoha ʻia o ʻelua mau helu.
Ka Hoʻopaʻa ʻana o Vector
ʻO ka vector \(\mathbf{v}\) i loko o kahi ʻōnaehana hoʻonohonoho 2D e hōʻike pinepine ʻia e like me \(\mathbf{v} = (v_x, v_y)\), kahi ʻo \(v_x\) a me \(v_y\) nā ʻāpana o ka vector ma nā axes x- a me y, kēlā me kēia. Ma ke ʻano hōʻailona ʻē aʻe, hiki ke kākau ʻia ka vector e like me \(\mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j}\), kahi ʻo \(\mathbf{i}\) a me \(\mathbf{j}\) nā vectors unit ma nā axes x- a me y, kēlā me kēia.
Vector Kūlana
ʻO ka vector kūlana kahi laʻana maʻalahi o kahi vector, i hoʻohana pinepine ʻia e hōʻike i ke kūlana o kahi kiko e pili ana i ke kumu. Inā aia ke kiko A ma nā hoʻonohonoho (a, b), a laila ua hōʻailona ʻia ka vector kūlana mai ke kumu a i ke kiko A ʻo \(\mathbf{A} = (a, b)\).
Hōʻike Kiʻi
Hiki ke hōʻike ʻia kahi vector ma ke ʻano he pua ma ka mokulele hoʻonohonoho me kona huelo ma ke kumu (0, 0) a me kona wēlau ma ke kiko (v_x, v_y). Hōʻike kēia pua i ka mamao a me ke kuhikuhi hea o ke kiko mai ke kumu.
Nā Hana Kumu ma nā Vectors
Hoʻohui Vector
Hoʻohui ʻia nā vectors ʻelua ma ka hoʻohui ʻana i kā lākou mau ʻāpana. No ka laʻana, inā loaʻa iā mākou ʻelua vectors \(\mathbf{u} = (u_x, u_y)\) a me \(\mathbf{v} = (v_x, v_y)\), a laila ʻo ka hoʻohui ʻana o kēia mau vectors ʻelua:
\[
\mathbf{u} + \mathbf{v} = (u_x + v_x, u_y + v_y)
\]
Ma ke ʻano geometric, hiki ke ʻike ʻia ka hopena o kēia hoʻohui ʻana ma ke kau ʻana i ka huelo o ka vector ʻelua ma ka piko o ka vector mua, a ʻo ka vector hopena ka vector e hoʻopili ana i ka huelo o ka vector mua me ka piko o ka vector ʻelua.
Ka Hoʻemi Vector
ʻO ka unuhi ʻana i ʻelua vectors ua like ia me ka hoʻohui ʻana iā lāua, akā ua unuhi ʻia nā ʻāpana o nā vectors. Inā loaʻa iā mākou nā vectors \(\mathbf{u}\) a me \(\mathbf{v}\) e like me luna, ʻo ka unuhi ʻana penei:
\[
\mathbf{u} – \mathbf{v} = (u_x – v_x, u_y – v_y)
\]
Hoʻonui Scala
ʻO ka hoʻonui scalar ka hana e hoʻonui ʻia ai kahi vector e kahi helu (scalar). Inā ʻo \(\mathbf{v} = (v_x, v_y)\) a he scalar ʻo k, a laila:
\[
k \mathbf{v} = (k v_x, k v_y)
\]
Huahana Kiko
ʻO ka huahana kiko o nā vectors ʻelua \(\mathbf{u}\) a me \(\mathbf{v}\) e hana i kahi scalar a ua hoʻokumu ʻia penei:
\[
\mathbf{u} \cdot \mathbf{v} = u_x v_x + u_y v_y
\]
Hāʻawi ka hopena o kēia hana i ka ʻike e pili ana i ka nui o nā ʻāpana o kēia mau vectors ʻelua ma ke ala like kekahi me kekahi.
Ka Lōʻihi (Ka Nui) o ka Vector
Hiki ke helu ʻia ka lōʻihi a i ʻole ka nui o ka vector \(\mathbf{v} = (v_x, v_y)\) me ka hoʻohana ʻana i ke ʻano hana:
\[
|\mathbf{v}| = \sqrt{v_x^2 + v_y^2}
\]
Hōʻike kēia lōʻihi i ka mamao mai ke kumu a i ke kiko (v_x, v_y) ma nā hoʻonohonoho Cartesian.
Nā noi Vector
ʻO ke kinoea
I ke ʻano physics, hoʻohana pinepine ʻia nā vectors e hōʻike i nā nui kino like ʻole e like me ka wikiwiki, ka wikiwiki, a me ka ikaika. No ka laʻana, inā neʻe kahi mea i kahi wikiwiki mau, i hōʻike ʻia e ka vector \(\mathbf{v}\), hiki ke helu ʻia ke ala i hele ʻia i kahi manawa i hāʻawi ʻia me ka hoʻohana ʻana i nā hana vector.
ʻEnekinia a me ka ʻenehana
I ke ʻenekinia, hoʻohana ʻia nā vectors no ka loiloi static a me ka dynamic o nā hale. No ka laʻana, hiki ke hōʻike ʻia nā ikaika e hana ana ma kahi hale ʻenekinia ma ke ʻano he vectors, a hana ʻia ka loiloi ma ka hōʻuluʻulu ʻana i nā vectors ikaika e ʻike i ka ikaika kū'ē e pono ai.
Nā Kiʻi Kamepiula
I nā kiʻi kamepiula, hoʻohana ʻia nā vectors e hōʻike i nā hoʻololi geometric like ʻole e like me ka unuhi, ka hoʻohuli ʻana, a me ka scaling. Hoʻohana pū ʻia nā vectors i ke kukui a me ka malu e hoʻoholo ai i ke kuhikuhi a me ka ikaika o ke kukui e pā ana i nā mea i loko o kahi hiʻohiʻona 3D.
ʻEkonometrika a me ka ʻEpekema ʻIkepili
I loko o ka econometrics a me ka ʻepekema ʻikepili, hoʻohana pinepine ʻia nā vectors i nā ʻano helu like ʻole a me ke aʻo mīkini. No ka laʻana, hoʻohana ʻia nā vectors ʻano hoʻokomo i nā algorithms aʻo mīkini e wānana a hoʻokaʻawale paha i ka ʻikepili.
Ka hopena
He mau mea hana ikaika nā vectors ʻelua-dimensional i nā ʻano aʻo like ʻole. ʻO ka ʻike kumu o ke ʻano o ka hōʻike ʻia ʻana o nā vectors a pehea e hana ʻia ai nā hana kumu ma luna o lākou he mea nui ia no kā lākou noi hou aku. Mai ka physics a i nā kiʻi kamepiula, a mai ka ʻenekinia a i ka ʻepekema ʻikepili, kōkua nā manaʻo vector iā mākou e hoʻomaopopo a hoʻohālike i ke ao a puni mākou ma ke ʻano ʻoi aku ka maikaʻi a me ka hoʻonohonoho ʻia. ʻO ka hoʻokele ʻana i kēia mau manaʻo e wehe i ka puka i ka nānā hou ʻana a me ka hoʻomohala ʻana ma nā ʻano like ʻole.
