Laʻana o nā nīnau kūkākūkā Vector: Nā Vectors like o ka helu like
He manaʻo nui nā vectors i ka makemakika a me ke kino. ʻOiai ke ʻano maʻalahi, he kuleana koʻikoʻi ko nā vectors i nā noi like ʻole, e like me ke ana ʻana o ka neʻe ʻana i ka kino, nā kiʻi kamepiula, a me ka nānā ʻana i ka ʻikepili i nā helu helu. Ma kēia ʻatikala, e kūkākūkā mākou i nā vectors, ʻo ia hoʻi nā vectors like, a hāʻawi i nā laʻana a me nā hoʻonā.
Ke Hoʻomaopopo nei i nā Vectors
ʻO ka vector kahi nui i loaʻa ka nui a me ke kuhikuhi. No ka laʻana, inā makemake ʻoe e hōʻike i kahi vector i loko o kahi mokulele ʻelua-dimensional, hiki iā ʻoe ke hoʻohana i ʻelua mau ʻāpana: hoʻokahi ma ka axis-x a hoʻokahi ma ka axis-y. I ka hōʻailona makemakika, hōʻike pinepine ʻia nā vectors e kahi pua ma luna o ka hōʻailona, e like me \(\vec{a}\), a i ʻole i kākau ʻia ma ka hōʻailona ʻāpana e like me \(\vec{a} = (a_x, a_y)\).
Ka Hoʻopaʻa ʻana o Vector
1. Ka Hoʻopaʻa ʻAno Geometric: ʻO ke ʻano geometric o kahi vector he ʻāpana laina i kuhikuhi ʻia me kahi kiko hoʻomaka a me kahi kiko hopena. Hōʻike ka lōʻihi o ka ʻāpana laina i ka nui (ka nui o ka vector), ʻoiai ke kuhikuhi o ka ʻāpana laina e hōʻike ana i ke kuhikuhi o ka vector.
2. Ka Hoʻopaʻa ʻĀpana: Ma kahi ākea ʻelua-dimensional, hiki ke hōʻike ʻia ka vector \(\vec{a}\) e like me \( \vec{a} = (a_x, a_y)\), kahi ʻo \(a_x\) ka ʻāpana o ka vector ma ke axis-X, a ʻo \(a_y\) ka ʻāpana o ka vector ma ke axis-Y.
3. Ka Hoʻomaopopo Kumu: Ma ka hakahaka ʻekolu-dimensional, hiki ke hōʻike ʻia ka vector \(\vec{b}\) e like me \( \vec{b} = b_x \hat{i} + b_y \hat{j} + b_z \hat{k} \), kahi ʻo \( \hat{i}, \hat{j}, \) a me \( \hat{k} \) he mau vectors unit ma nā axes X, Y, a me Z.
Ke kūlike o ka Vector
Ua ʻōlelo ʻia ʻelua mau vectors he like inā like ko lākou nui a me ke kuhikuhi, me ka nānā ʻole i ko lākou kūlana ma ka lewa. No ka laʻana, inā like nā ʻāpana o nā vectors \(\vec{a}\) a me \(\vec{b}\), a laila ua like lākou:
\[
\vec{a} = \vec{b} \iff a_x = b_x \text{ a me } a_y = b_y \text{ ma 2D}
\]
\[
\vec{a} = \vec{b} \iff a_x = b_x, a_y = b_y, \text{ a me } a_z = b_z \text{ ma 3D}
\]
Nā Nīnau Laʻana a me ke Kūkākūkā
Eia kekahi mau laʻana o nā nīnau a me nā kūkākūkā e pili ana i nā vectors like.
Laʻana 1: Ke hōʻoia nei i ke kūlike o ka Vector 2D
Nīnau: Hāʻawi ʻia i ʻelua mau vectors ma ka mokulele ʻelua-dimensional, \(\vec{u} = (3, 4)\) a me \(\vec{v} = (3, 4)\). Ua like anei kēia mau vectors ʻelua?
Kūkākūkā:
No ka hōʻoia ʻana inā like kēia mau vectors ʻelua, pono mākou e hōʻoia ua like nā ʻāpana pili o nā vectors:
– ʻO ka ʻāpana \( x \) o \(\vec{u}\) he 3, a ʻo ka ʻāpana \( x \) o \(\vec{v}\) he 3 nō hoʻi.
– ʻO ka ʻāpana \( y \) o \(\vec{u}\) he 4, a ʻo ka ʻāpana \( y \) o \(\vec{v}\) he 4 nō hoʻi.
No ka mea, ua like ʻo \( u_x = v_x \) a me \( u_y = v_y \), a laila ua like ʻo \(\vec{u}\) a me \(\vec{v}\). No laila, ua like ʻo \(\vec{u} = \vec{v}\).
Laʻana 2: Ke hōʻoia nei i ke kūlike o ka Vector 3D
Nīnau: Hāʻawi ʻia i ʻelua mau vectors i loko o kahi ʻekolu-dimensional, \(\vec{a} = (1, -2, 3)\) a me \(\vec{b} = (1, -2, 3)\). Ua like anei kēia mau vectors ʻelua?
Kūkākūkā:
No ka hōʻoia ʻana i ke kūlike ma ka hakahaka ʻekolu-dimensional, ke nānā pū nei mākou i nā ʻāpana pākahi:
– ʻO ka ʻāpana \( x \) o \(\vec{a}\) he 1, a ʻo ka ʻāpana \( x \) o \(\vec{b}\) he 1 nō hoʻi.
– ʻO ka ʻāpana \( y \) o \(\vec{a}\) he -2, a ʻo ka ʻāpana \( y \) o \(\vec{b}\) he -2 nō hoʻi.
– ʻO ka ʻāpana \( z \) o \(\vec{a}\) he 3, a ʻo ka ʻāpana \( z \) o \(\vec{b}\) he 3 nō hoʻi.
No ka mea, ua like ʻo \( a_x = b_x \), \( a_y = b_y \), a me \( a_z = b_z \), a laila ua like ʻo \(\vec{a}\) a me \(\vec{b}\). No laila, ua like ʻo \(\vec{a} = \vec{b}\).
Laʻana 3: Nā Vectors ʻAʻole like
Nīnau: Hāʻawi ʻia i ʻelua mau vectors \(\vec{p} = (2, 4)\) a me \(\vec{q} = (3, 4)\). Ua like anei kēia mau vectors ʻelua?
Kūkākūkā:
No ka nānā ʻana i ke kaulike, nānā mākou i nā ʻāpana o nā vectors ʻelua:
– ʻO ka ʻāpana \( x \) o \(\vec{p}\) he 2, ʻoiai ʻo ka ʻāpana \( x \) o \(\vec{q}\) he 3. Ua maopopo ʻaʻole like nā ʻāpana \( x \).
– ʻO ka ʻāpana \( y \) o \(\vec{p}\) he 4, a ʻo ka ʻāpana \( y \) o \(\vec{q}\) he 4 nō hoʻi.
ʻOiai hoʻokahi wale nō ʻāpana i like ʻole (\( p_x \neq q_x \)), ʻaʻole like nā vectors ʻelua. No laila, \(\vec{p} \neq \vec{q}\).
Laʻana 4: Ka nui o ka Vector
Nīnau: Hāʻawi ʻia i ʻelua mau vectors i loko o kahi ākea ʻelua-dimensional, \(\vec{m} = (2, 6)\) a me \(\vec{n} = (4, 3)\). Ua like anei kēia mau vectors ʻelua i ka nui?
Kūkākūkā:
ʻO ka hana mua, ʻo ia ka helu ʻana i nā nui o nā vectors ʻelua. ʻO ka nui o kahi vector \(\vec{v} = (v_x, v_y)\) ma nā ana ʻelua:
\[
||\vec{v}|| = \sqrt{v_x^2 + v_y^2}
\]
No ke vector \(\vec{m} = (2, 6)\):
\[
||\vec{m}|| = \sqrt{2^2 + 6^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10}
\]
No ke vector \(\vec{n} = (4, 3)\):
\[
||\vec{n}|| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]
ʻOiai ʻo \(||\vec{m}|| \neq ||\vec{n}||\), ʻaʻole like kēia mau vectors ʻelua ma ke ʻano o ko lākou nui.
Ka hopena
He mea koʻikoʻi ka hoʻomaopopo ʻana i ke kumumanaʻo o nā vectors, ʻoiai nā vectors like, i nā noi like ʻole o ka makemakika a me ke kino. Loaʻa i nā vectors like nā ʻāpana like ma kēlā me kēia axis, me ka nānā ʻole i ko lākou kūlana ma ka lewa. Me ka hoʻomaʻamaʻa lawa ma o nā laʻana a me nā kūkākūkā ʻana, hiki iā mākou ke hoʻoikaika i ko mākou ʻike i kēia kumumanaʻo a hoʻohana iā ia i nā kūlana like ʻole.
ʻO ka pahuhopu o kēia ʻatikala e hāʻawi i ka poʻe heluhelu i kahi ʻike maikaʻi aʻe i ke ʻano o ka nānā ʻana i ke kaulike ma waena o nā vectors ʻelua a pehea e hoʻopili ai i kēia manaʻo i nā pilikia like ʻole. ʻO ka ʻike ʻana he like nā vectors ʻelua e hiki ai iā mākou ke huki i nā hopena like ʻole i ka loiloi vector paʻakikī a me nā kahua ʻē aʻe.
