Nā Vectors i ka Physics: Nā Manaʻo a me nā Noi
He manaʻo makemakika koʻikoʻi nā vectors i ka physics. ʻAʻole e like me nā scalars, nona ka nui (a i ʻole ka waiwai) wale nō, loaʻa i nā vectors ka nui a me ke kuhikuhi. ʻO ka hoʻomaopopo piha ʻana i nā vectors a me kā lākou noi i ka physics hiki ke kōkua iā mākou e wehewehe i nā hanana kūlohelohe me ka pololei a me ka naʻauao.
Ke Hoʻomaopopo nei i nā Vectors
I ka ʻōlelo maʻalahi, he mea makemakika ka vector i hōʻike ʻia e kahi pua e hōʻike ana i kahi kuhikuhi a he lōʻihi kikoʻī e hōʻike ana i kona nui. Ua kapa pinepine ʻia kēia lōʻihi ʻo ka "magnitude" a i ʻole ka nui vector.
I mea e hoʻomaopopo maikaʻi ai i kēia manaʻo, e noʻonoʻo e hele wāwae mai kahi A a i kahi B. ʻO ka mamao ma waena o kēia mau wahi ʻelua he nui scalar, akā ʻo ke kuhikuhi o ka huakaʻi mai A a i B e hoʻolauna i ke kumumanaʻo o kahi vector. ʻO kou kūlana hope loa e pili ana i kou wahi hoʻomaka ʻaʻole ia e pili wale i ka mamao o kou hele ʻana akā i ke kuhikuhi āu e hele ai.
Hōʻike Vector
I ka hōʻailona makemakika, kākau pinepine ʻia nā vectors me nā leka wiwo ʻole e like me v, a i ʻole he mau leka me kahi pua ma luna o lākou e like me \( \vec{v} \). Hiki ke hōʻike ʻia nā vectors ʻelua-dimensional e like me nā hui i hoʻonohonoho ʻia \( (v_x, v_y) \), kahi ʻo \( v_x \) a me \( v_y \) nā ʻāpana o ka vector ma nā kuhikuhi x- a me y.
No kahi vector ʻekolu-dimensional, lilo ka hōʻike ʻana i \((v_x, v_y, v_z) \). Loaʻa pinepine ʻia kēia mau ʻāpana ma o ka projection o ka vector mua ma luna o nā axis x, y, a me z, a hiki ke hōʻike ʻia ma kahi ʻōnaehana hoʻonohonoho Cartesian.
Nā Hana ma nā Vectors
Hoʻohui a me ka Hoʻemi Vector
Hana ʻia ka hoʻohui vector ma ka hoʻohui ʻana i kā lākou mau ʻāpana. No ka laʻana, inā \(\vec{u} = (u_x, u_y, u_z)\) a me \(\vec{v} = (v_x, v_y, v_z)\), a laila ʻo ka huina vector \(\vec{w} = \vec{u} + \vec{v}\) penei:
\[
\vec{w} = (u_x + v_x, u_y + v_y, u_z + v_z)
\]
Hana ʻia ka unuhi vector ma ke ʻano like, ʻo ia hoʻi ma ka unuhi ʻana i kāna mau ʻāpana. No laila, ʻo \(\vec{w} = \vec{u} – \vec{v}\) penei:
\[
\vec{w} = (u_x – v_x, u_y – v_y, u_z – v_z)
\]
Hoʻonui ʻia o nā Vectors e nā Scalars
Hana ʻia ka hoʻonui ʻana o kahi vector e kahi scalar \(k\) ma ka hoʻonui ʻana i kēlā me kēia ʻāpana o ka vector e kēlā scalar. No ka laʻana, inā he scalar ʻo \(k\) a ʻo \(\vec{v} = (v_x, v_y, v_z)\), a laila ʻo ka hopena o ka hoʻonui ʻana i kahi vector e kahi scalar penei:
\[
k \cdot \vec{v} = (k v_x, k v_y, k v_z)
\]
Hoʻonui Vector (Huahana Kiko a me ka Huahana Keʻa)
ʻElua ʻano nui o ka hoʻonui vector, ʻo ia hoʻi ka huahana kiko a me ka huahana keʻa.
ʻO ka huahana kiko kahi hana e hoʻopuka ai i kahi huahana scalar. No ka laʻana, no nā vectors \(\vec{u}\) a me \(\vec{v}\):
\[
\vec{u} \cdot \vec{v} = u_x v_x + u_y v_y + u_z v_z
\]
Hoʻohana ʻia kēia hana i nā ʻano hana like ʻole, me ka hoʻoholo ʻana i ka projection o kekahi vector i kekahi.
ʻO ka huahana kea he hana e hana ana i kahi vector hou e kū pololei ana i nā vectors mua ʻelua. No ka laʻana, no nā vectors \(\vec{u}\) a me \(\vec{v}\):
\[
\vec{u} \times \vec{v} = (u_y v_z – u_z v_y, u_z v_x – u_x v_z, u_x v_y – u_y v_x)
\]
Hoʻohana nui ʻia nā huahana kea i ka physics, ʻoiai i nā hihia e pili ana i nā wili a me nā manawa o ka ikaika.
Nā noi o nā Vectors i ka Physics
ʻO ka hana aʻe, ʻo ia ke ʻike pehea e hoʻopili ʻia ai kēia mau manaʻo i ka physics.
Kinematika
I loko o ke kinematics, hiki ke hōʻike ʻia ke kūlana, ka wikiwiki, a me ka wikiwiki ma ke ʻano he mau vectors. Hiki ke hōʻike ʻia ke kūlana o kahi mea ma kahi kiko \(t \) ma ke ʻano he vector kūlana \(\vec{r}(t)\). ʻO ka wikiwiki, ʻo ia ka wikiwiki o ka loli o ke kūlana e pili ana i ka manawa, ʻo ia ka derivative mua o ke kūlana e pili ana i ka manawa:
\[
\vec{v}(t) = \frac{d\vec{r}(t)}{dt}
\]
ʻO ka hoʻolalelale ʻana, ʻo ia hoʻi ka wikiwiki o ka loli o ka wikiwiki e pili ana i ka manawa, ʻo ia ka derivative mua o ka wikiwiki a i ʻole ka derivative ʻelua o ke kūlana:
\[
\vec{a}(t) = \frac{d\vec{v}(t)}{dt} = \frac{d^2\vec{r}(t)}{dt^2}
\]
Hoʻoikaika kino
I loko o ka dynamics, he mea nui ka hoʻohana ʻana i nā vectors. ʻO ke kānāwai ʻelua o Newton, no ka laʻana, ke ʻōlelo nei ʻo ka ikaika e hana ana ma luna o kahi mea ʻo ia ka huahana o kona nuipa \(m\) a me kona wikiwiki \( \vec{a} \):
\[
\vec{F} = m \vec{a}
\]
Ma kēia hihia, ʻo ka ikaika \(\vec{F}\) a me ka wikiwiki \(\vec{a}\) he mau vectors. ʻO ke ʻano kēia, pono e noʻonoʻo ka nānā ʻana i nā ikaika ma kahi mea i ke ʻano vector o kēia mau ikaika.
Nā kahua uila a me nā kahua makeneka
Ma ke kahua o ka electromagnetism, ʻo ke kahua uila \(\vec{E}\) a me ke kahua magnetic \(\vec{B}\) he mau vectors nō hoʻi. Hoʻoholo ʻia ka ikaika a me ke kuhikuhi o ke kahua uila ma kekahi wahi ma ka lewa e ka vector kahua uila \(\vec{E}\), ʻoiai ke kahua magnetic \(\vec{B}\) e wehewehe ana i ke ʻano o ka hopena o ke kahua magnetic i ka lewa a puni. Hoʻopilikia lākou i nā ʻāpana i hoʻopiʻi ʻia ma nā ʻano hiki ke wānana ʻia e nā kānāwai o ka physics e hoʻohana ana i nā vectors.
Nā Mekanika Wai
I loko o ka mechanics fluid, hoʻohana ʻia nā vectors e wehewehe i ka wikiwiki a me ka vorticity o kahi wai. ʻO ka wikiwiki o ka wai \(\vec{v}(\vec{r}, t)\) ma ke kūlana \(\vec{r}\) a me ka manawa \(t\) he vector. He paʻakikī loa ka mechanics fluid a hilinaʻi pinepine ʻia i nā kaulike Navier-Stokes, ʻo ia hoʻi nā kaulike vector partial differential.
Nā Optics a me nā Nalu
I loko o ke kinoea nalu, ʻoi aku hoʻi i nā kaapuni electromagnetic, hiki ke hōʻike ʻia nā kahua uila a me nā kahua magnetic e hoʻomalu ana i ka laha ʻana o ka nalu ma ke ʻano he vectors. ʻO nā hanana e like me ka polarization o ke kukui e pili ana i ka nānā ʻana o ka vector o ke kahua uila no ka mea ʻo ke kuhikuhi o ke kahua uila e hoʻoholo ai i ka polarization.
Ka pilina
Ma ke kumumanaʻo o ka relativity, ua hoʻonui ʻia ke kumumanaʻo o nā vectors e hoʻokomo i ke kumumanaʻo o "four-vectors," ʻo ia hoʻi ka nui o ka manawa a me ʻekolu mau nui o ka spatial. ʻO kahi laʻana o kahi vector ʻehā ʻo ia ka "four-momentum," kahi e hoʻohui pū ai i ka ikehu a me ka momentum i loko o hoʻokahi vector entity.
Ka hopena
ʻO nā vectors kekahi o nā manaʻo makemakika koʻikoʻi loa i ka physics. Hāʻawi lākou i kahi ala ʻoi aku ka maʻalahi a me ka pololei e hoʻohālike i nā hanana kūlohelohe, ʻoiai ke kuhikuhi ʻana he mea nui. Mai ke kinematics a i ke kumumanaʻo o ka relativity, ʻo ka hoʻohana ʻana i nā vectors e maʻalahi ai ka hoʻomaopopo ʻana, ka nānā ʻana, a me ka wānana ʻana i nā ʻano hanana kino like ʻole. No laila, ʻo ka hoʻopaʻa ʻana i ke kumumanaʻo o nā vectors he ʻanuʻu koʻikoʻi ia no ka poʻe makemake e aʻo i ka physics.
