Nā Huaʻōlelo a me nā ʻAno o ka Hoʻopaʻa ʻana o Vector
I ke kūkākūkā ʻana i ka makemakika, ka physics, a me ka ʻepekema kamepiula, ʻo ke kumumanaʻo o nā vectors he mea koʻikoʻi pinepine ia e hoʻomaopopo. ʻAʻole wale nā vectors he mau manaʻo abstract; pili lākou i nā ʻano hana like ʻole, e like me ka nānā ʻana i ka ʻikepili, nā kiʻi kamepiula, a me nā simulations physics. Ma kēia ʻatikala, e kūkākūkā mākou i ka ʻōlelo vector a me ka notation, a laila e ʻimi i nā ʻano vectors like ʻole i loaʻa i kēia mau aʻo.
Nā Huaʻōlelo Vector a me ka Hoʻopaʻa inoa
1. Nā Vectors a me nā Scalars
He mea makemakika ka vector nona ka nui a me ke kuhikuhi. I ka hoʻohālikelike ʻana, he waiwai hoʻokahi ka scalar nona ka nui wale nō a ʻaʻohe kuhikuhi. No ka laʻana, ʻo ka wikiwiki o 5 m/s me ka ʻole o ke kuhikuhi ʻana he scalar, ʻoiai ʻo ka wikiwiki o 5 m/s hikina he vector.
2. Ka Hoʻopaʻa ʻana o ka Vector
Hoʻike pinepine ʻia nā vectors e kahi leka liʻiliʻi wiwo ʻole e like me v , a i ʻole e kahi pua ma luna o ka leka e like me \(\vec{v}\). No ka laʻana, inā loaʻa iā mākou kahi vector v nona nā ʻāpana \(v_1, v_2, v_3\), a laila hiki ke kākau ʻia penei:
\[ \vec{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} \]
ʻO kekahi ala e kākau ai i nā vectors, ʻoi aku hoʻi i nā pōʻaiapili ʻelua a ʻekolu paha, ʻo ia ka hoʻohana ʻana i kahi kumu maʻamau. Eia kekahi laʻana:
\[ \vec{v} = v_1\hat{i} + v_2\hat{j} + v_3\hat{k} \]
kahi ʻo \(\hat{i}, \hat{j}\), a me \(\hat{k}\) he mau vectors unit ma nā axis x, y, a me z.
Nā ʻAno o nā Vectors
1. Vector Kūlana
ʻO ka vector kūlana he vector e wehewehe ana i ke kūlana o kahi kiko ma ka lewa e pili ana i kahi kiko kuhikuhi, ʻo ia hoʻi ke kiko O (ke kumu). Inā loaʻa i ka kiko P nā hoʻonohonoho (x, y, z) ma ka lewa 3D, a laila hiki ke hōʻike ʻia ka vector kūlana \(\vec{r}\) penei:
\[ \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \]
2. Vector Neʻe
Hōʻike ka vector neʻe ʻana i ka loli o ke kūlana o kahi kiko mai kekahi kūlana a i kekahi. Manaʻo ʻia he mau hoʻonohonoho ko ka kiko A (x1, y1, z1) a he mau hoʻonohonoho ko ka kiko B (x2, y2, z2). Hiki ke kākau ʻia ka vector neʻe ʻana \(\vec{d}\) mai A a i B penei:
\[ \vec{d} = (x2 – x1)\papale{i} + (y2 – y1)\papale{j} + (z2 – z1)\papae{k} \]
3. Vector wikiwiki
ʻO ka wikiwiki kahi vector e hōʻike ana i ka wikiwiki o ka loli o ke kūlana o kahi mea i kēlā me kēia manawa. Inā ʻo \(\vec{r}(t)\) kahi hana o ke kūlana e pili ana i ka manawa, ʻo ka vector wikiwiki \(\vec{v}(t)\) ka derivative o \(\vec{r}(t)\) e pili ana i ka manawa t:
\[ \vec{v}(t) = \frac{d\vec{r}(t)}{dt} \]
4. Vector Hoʻolalelale
ʻO ka vector wikiwiki ka derivative o ka vector wikiwiki e pili ana i ka manawa. Hōʻike ia i ka wikiwiki o ka loli o ka wikiwiki o kahi mea i kēlā me kēia manawa. Inā ʻo \(\vec{v}(t)\) kahi hana o ka wikiwiki e pili ana i ka manawa, ʻo ka vector wikiwiki \(\vec{a}(t)\) ka derivative o \(\vec{v}(t)\):
\[ \vec{a}(t) = \frac{d\vec{v}(t)}{dt} \]
5. Hoʻoikaika Vector
Wahi a ke kānāwai ʻelua o Newton, ʻo ka ikaika ka huahana o ka nuipa a me ka wikiwiki. He vector nō hoʻi ka ikaika no ka mea he nui a he kuhikuhi kona. Inā ʻo m ka nuipa a ʻo \(\vec{a}\) ka vector wikiwiki, a laila hiki ke hōʻike ʻia ka vector ikaika \(\vec{F}\) penei:
\[ \vec{F} = m\vec{a} \]
6. Vector Unit
ʻO ka vector unit he vector nona ka nui (lōʻihi) o hoʻokahi. Hiki ke loaʻa ka vector unit o kahi vector \(\vec{v}\) ma ka puʻunaue ʻana iā \(\vec{v}\) me kona nui. Inā he nui ko \(\vec{v}\) o \(||\vec{v}||\), a laila hiki ke kākau ʻia kona vector unit penei:
\[ \hat{v} = \frac{\vec{v}}{||\vec{v}||} \]
7. ʻAʻohe Vector
ʻO ka vector zero kahi vector nona nā ʻāpana āpau he zero, a hōʻike pinepine ʻia e \(\vec{0}\). ʻAʻohe kuhikuhi o kēia vector a ʻo kona nui he zero. ʻO kahi laʻana ma ka hakahaka ʻekolu-dimensional penei:
\[ \vec{0} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \]
8. Nā Vectors Orthogonal
Ua ʻōlelo ʻia ʻelua mau vectors he orthogonal inā ʻaʻohe o lākou huahana kūloko. Inā ʻelua mau vectors ʻo \(\vec{u}\) a me \(\vec{v}\), a laila he orthogonal lāua inā:
\[ \vec{u} \cdot \vec{v} = 0 \]
9. Nā Vectors Collinear a me nā Vectors Coplanar
Ua ʻōlelo ʻia ʻelua mau vectors he collinear inā e moe ana lākou ma ka laina pololei like a i ʻole e like ana. Hiki ke hōʻike ʻia lākou ma ke ʻano he mau scalar multiples o kekahi i kekahi. Eia kekahi laʻana:
\[ \vec{v} = k\vec{u} \]
no kekahi scalar \(k\).
I kēia manawa, ua ʻōlelo ʻia ʻekolu mau vectors he coplanar inā lākou e moe ana ma ka mokulele like. Hiki ke hōʻike ʻia lākou ma ke ʻano he hui linear o nā vectors ʻelua ʻē aʻe.
Nā Hana ma nā Vectors
1. Hoʻohui a me ka Hoʻemi Vector
Hana ʻia ka hoʻohui vector ma ka hoʻohui ʻana i kā lākou mau ʻāpana pili. Inā \(\vec{u} = \begin{pmatrix} u_1 \\ u_2 \\ u_3 \end{pmatrix}\) a me \(\vec{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}\), a laila:
\[ \vec{u} + \vec{v} = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \\ u_3 + v_3 \end{pmatrix} \]
Hana ʻia ka unuhi ʻana ma ka unuhi ʻana i nā ʻāpana like:
\[ \vec{u} – \vec{v} = \begin{pmatrix} u_1 – v_1 \\ u_2 – v_2 \\ u_3 – v_3 \end{pmatrix} \]
2. Hoʻonui Scalalar
ʻO ka hoʻonui scalar kahi hana e pili ana i kahi vector me kahi scalar (waiwai helu). Inā he scalar ʻo k a \(\vec{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}\), a laila:
\[ k\vec{v} = \begin{pmatrix} kv_1 \\ kv_2 \\ kv_3 \end{pmatrix} \]
3. Huahana Kūloko (Huahana Kiko)
He scalar ka huahana kūloko o nā vectors ʻelua \(\vec{u}\) a me \(\vec{v}\). Hiki ke helu ʻia ma o:
\[ \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3 \]
4. Huahana Kea
ʻO ka huahana kea o nā vectors ʻelua \(\vec{u}\) a me \(\vec{v}\) e hoʻopuka i kahi vector hou e orthogonal i kēia mau vectors ʻelua. Ma ka hakahaka ʻekolu-dimensional, ua helu ʻia kēia penei:
\[ \vec{u} \manawa \vec{v} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
u_1 a me u_2 a me u_3 \\
v_1 a me v_2 a me v_3
\end{vmatrix} \]
Ka hopena
He mea koʻikoʻi ka hoʻomaopopo ʻana i ka ʻōlelo vector a me ka notation, a me ko lākou ʻano, i nā ʻano aʻo ʻepekema like ʻole. ʻAʻole wale nā Vectors he mau mea makemakika abstract, akā he mau mea hana ikaika hoʻi i ka physics, ka ʻenekinia, a me ka loiloi ʻenehana ʻike. Me ka hoʻomaopopo maikaʻi ʻana i kēia mau manaʻo kumu, hiki iā mākou ke hoʻoponopono maʻalahi i nā pilikia paʻakikī ma nā ʻano kahua like ʻole.
