Nā Vector Kūlana: Nā Kumu, Nā Hoʻohana, a me nā Laʻana i ke Ola o kēlā me kēia Lā
Pendahuluan
He manaʻo koʻikoʻi ka vector kūlana i ka makemakika a me ke kino i hoʻohana ʻia e wehewehe i ke kūlana o kahi kiko ma ka lewa. I ka ʻōlelo maʻalahi, hiki ke manaʻo ʻia kahi vector kūlana he pua e kuhikuhi ana mai kahi hoʻomaka (ʻo ke kumu maʻamau) i kahi kiko i kuhikuhi ʻia. E kūkākūkā kēia ʻatikala i ka wehewehe ʻana o kahi vector kūlana, kona mau ʻāpana, pehea e helu ai, a me kāna mau noi pono i ke ola o kēlā me kēia lā.
Ka Wehewehena o ke Vector Kūlana
ʻO ka vector kūlana kahi vector e hoʻopili i ke kumu i kahi kiko ma ka lewa. Ke hana nei i ka lewa ʻelua-dimensional (2D), ua hōʻike ʻia kahi vector kūlana ma ke ʻano he hui i kauoha ʻia \((x, y)\), kahi \(x\) a me \(y\) nā hoʻonohonoho o ke kiko. Ma ka lewa ʻekolu-dimensional (3D), ua hōʻike ʻia kahi vector kūlana ma ke ʻano he triplet i kauoha ʻia \((x, y, z)\).
Eia kekahi laʻana, inā loaʻa iā kākou kahi kiko A ma nā hoʻonohonoho (3, 4) ma kahi ākea 2D, a laila ʻo ka vector kūlana e hoʻopili ana i ke kumu (0,0) i ke kiko A ʻo \(\mathbf{r} = 3\mathbf{i} + 4\mathbf{j}\), kahi ʻo \(\mathbf{i}\) a me \(\mathbf{j}\) he mau vectors unit ma nā axis \(x\) a me \(y\).
Nā ʻāpana Vector Kūlana
ʻO ka vector kūlana he mau ʻāpana e hōʻike ana i nā mamao ma nā axis coordinate. Ma ka hakahaka 2D, hiki ke hōʻike ʻia ka vector kūlana \(\mathbf{r}\) penei:
\[
\mathbf{r} = x\mathbf{i} + y\mathbf{j}
\]
Maanei, ʻo \(x\) ka ʻāpana o ka vector kūlana ma ke axis \(x\), a ʻo \(y\) ka ʻāpana o ka vector kūlana ma ke axis \(y\).
Ma ka lewa 3D, ua hōʻike ʻia ka vector kūlana \(\mathbf{r}\) penei:
\[
\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}
\]
Maanei, ʻo \(x\), \(y\), a me \(z\) nā ʻāpana o ka vector kūlana ma nā axes \(x\), \(y\), a me \(z\), kēlā me kēia, ʻoiai ʻo \(\mathbf{k}\) ka vector unit ma ke axis \(z\).
Pehea e helu ai i ke kūlana Vector
ʻO ka helu ʻana i kahi vector kūlana e pili ana i ka hoʻoholo ʻana i ka mamao mai ke kumu a i ke kiko i nīnau ʻia ma nā hoʻonohonoho Cartesian. No ka laʻana, inā aia ke kiko B ma nā hoʻonohonoho (5, 7) ma ka hakahaka 2D, ʻo ke vector kūlana e hoʻopili ana i ke kumu (0, 0) i ke kiko B penei:
\[
\mathbf{r_B} = 5\mathbf{i} + 7\mathbf{j}
\]
No ka helu ʻana i ka lōʻihi o ka vector kūlana (a i ʻole ka nui), hoʻohana mākou i ka Pythagorean theorem. Hāʻawi ʻia ka lōʻihi o ka vector kūlana \(\mathbf{r}\) ma ka hakahaka 2D e:
\[
|\mathbf{r}| = \sqrt{x^2 + y^2}
\]
Ma ka hakahaka 3D, ua helu ʻia ka lōʻihi o ka vector kūlana \(\mathbf{r}\) penei:
\[
|\mathbf{r}| = \sqrt{x^2 + y^2 + z^2}
\]
No ka laʻana, no ke kiko C ma nā hoʻonohonoho (3, 4, 5) ma ka hakahaka 3D, ʻo ka lōʻihi o ka vector kūlana \(\mathbf{r_C}\) penei:
\[
|\mathbf{r_C}| = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} \approx 7.07
\]
Nā Hoʻohana o nā Vector Kūlana i ke Ola o kēlā me kēia Lā
He nui nā hoʻohana pono o nā vectors kūlana ma nā ʻano like ʻole. Eia kekahi mau laʻana o kā lākou hoʻohana ʻana i ke ola o kēlā me kēia lā:
1. Hoʻokele a me GPS
Ma nā ʻōnaehana hoʻokele e like me GPS, hoʻohana ʻia nā vectors kūlana e hoʻoholo ai i ka wahi o ka mea hoʻohana e pili ana i nā satelite GPS. A laila hoʻohana ʻia kēia ʻikepili kūlana e helu i ka mamao a me ke kuhikuhi o ka huakaʻi.
2. ʻEnekinia Kīwila a me ke Kuhikuhipuʻuone
Hoʻohana nā ʻenekinia kīwila a me nā mea kākau hale i nā vectors kūlana e hoʻolālā i nā hale a me nā pono hana. Kōkua kēia mau vectors i ka hoʻoholo ʻana i nā kūlana pili o nā ʻano mea kūkulu like ʻole.
3. Ka Hōkū Kilokilo
I ka ʻike hōkū, hoʻohana ʻia nā vector kūlana e wehewehe i nā kūlana o nā hōkū, nā hōkūhele, a me nā mea lani ʻē aʻe e pili ana i ka Honua a i ʻole ke kikowaena o ka ʻōnaehana lā.
4. Kino
I ke ʻano physics, he mea nui nā vectors kūlana no ka nānā ʻana i ka neʻe ʻana. No ka laʻana, i ka nānā ʻana i ka neʻe ʻana o ka projectile, hoʻohana ʻia nā vectors kūlana e hoʻoholo ai i ke kūlana o kahi mea i nā manawa like ʻole.
5. Nā Lopako
I loko o ka robotics, hoʻohana ʻia nā vectors kūlana e kāohi i ka neʻe ʻana o kahi robot. Kōkua kēia mau vectors i ka robot e hoʻoholo i kona kūlana pololei a me ke kuhikuhi o ka neʻe ʻana.
Nā Nīnau a me nā Hoʻonā Laʻana
Eia kekahi laʻana o ka nīnau e hoʻomālamalama ai i ka ʻike ʻana i nā vectors kūlana.
Nīnau:
Aia ke kiko P ma nā hoʻonohonoho (2, 3) a aia ke kiko Q ma nā hoʻonohonoho (5, 7) ma kahi ākea 2D. E hoʻoholo i ka vector kūlana e hoʻopili ana i ke kiko P i ke kiko Q a helu i ka lōʻihi o ka vector.
Hoʻonā:
ʻO ke vector kūlana \(\mathbf{PQ}\) ka vector e hoʻopili ana i ke kiko P i ke kiko Q. Hiki iā mākou ke helu i nā ʻāpana o ka vector \(\mathbf{PQ}\) ma ka unuhi ʻana i nā hoʻonohonoho o P mai nā hoʻonohonoho o Q:
\[
\mathbf{PQ} = (5 – 2)\mathbf{i} + (7 – 3)\mathbf{j} = 3\mathbf{i} + 4\mathbf{j}
\]
ʻO ka lōʻihi o ka vector kūlana \(\mathbf{PQ}\):
\[
|\mathbf{PQ}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
No laila, ʻo ke vector kūlana e hoʻohui ana i ka kiko P i ka kiko Q ʻo \(3\mathbf{i} + 4\mathbf{j}\) a ʻo ka lōʻihi o ka vector he 5 mau ʻāpana.
Ka hopena
ʻO ka vector kūlana kahi manaʻo nui i hoʻohana ʻia e wehewehe i kahi o kahi kiko ma ka lewa. ʻO ka hoʻomaopopo ʻana i nā ʻāpana a pehea e helu ai i kahi vector kūlana he mea nui ia no nā ʻano hana like ʻole, mai ka hoʻokele ʻana a i ka nānā kino. ʻO ka hoʻomaopopo ʻana i kēia mau kumu e maʻalahi ai ka hoʻomaopopo a me ka hoʻopili ʻana i ke kumumanaʻo o nā vector kūlana i ke ola o kēlā me kēia lā a me nā ʻoihana ʻoihana.
