Nā nīnau hoʻohālike e kūkākūkā ana i ka huaʻōlelo, ka hōʻailona, a me nā ʻano vectors
He mea nui ka hoʻomaopopo ʻana i nā vectors a me kona ʻike ʻana i nā lālā like ʻole o ka ʻepekema, ʻoiai ke kino a me ka makemakika. Hiki i ka hoʻohana pono ʻana i nā vectors ke kōkua i ka nānā ʻana i nā pilikia a me ka loaʻa ʻana o nā hopena kūpono. Kūkākūkā kēia ʻatikala i nā huaʻōlelo a me nā hoʻomaopopo e pili ana i nā vectors, e hōʻike ana iā lākou me nā laʻana a me nā wehewehe piha.
Nā Huaʻōlelo Vector
No ka hoʻomaopopo ʻana i nā vectors, pono mākou e hoʻomaopopo mua i nā huaʻōlelo maʻamau:
1. Vector: He nui nona ka nui (waiwai nui) a me ke kuhikuhi. Hōʻailona pinepine ʻia nā vectors e nā leka wiwo ʻole e like me A, a, a i ʻole e kahi hōʻailona pua ma luna o lākou e like me \(\vec{A}\).
2. Ka nui (Waiwai Nui): ʻO kēia ka lōʻihi a i ʻole ka nui o ka vector. Ua hōʻike ʻia e | A | a i ʻole \(\|\vec{A}\|\).
3. Ke Poʻo a me ka Huelo: Ma ka hōʻike kiʻi, ua hōʻike ʻia nā vectors ma ke ʻano he mau pua. Ua kapa ʻia ke kiko hoʻomaka o ka pua ʻo ka huelo a ua kapa ʻia ke kiko hope o ka pua ʻo ke poʻo.
4. Nā Vectors like: Nā vectors e like ana kekahi me kekahi a i ʻole ma ka laina hana like.
5. Nā Veka Collinear: Nā veka e moe ana ma kahi laina pololei hoʻokahi.
6. Vector Resultant: He vector hoʻokahi i loaʻa ka hopena like me ka hopena i hui pū ʻia o ʻelua a ʻoi aku paha nā vectors.
Ka Hoʻopaʻa ʻana o Vector
He nui nā lula o ka hōʻailona vector e pono e hoʻomaopopo ʻia e wehewehe pono a kākau i nā vectors.
1. Ka Hoʻailona Hua Moa a me ka Pua: Hōʻailona pinepine ʻia nā Vectors me nā hua moa a i ʻole nā pua. Nā Laʻana: A, B, a i ʻole \(\vec{A}\).
2. Nā Hoʻonohonoho Vector: Ua hōʻailona ʻia nā Vectors ma kahi ʻelua-dimensional (2D) ʻo \(\vec{A} = (A_x, A_y)\), ʻoiai ma kahi ʻekolu-dimensional (3D) ua hōʻailona ʻia lākou ʻo \(\vec{A} = (A_x, A_y, A_z)\).
3. Nā Vector Kumu: Ma ka lewa 2D a me 3D, ʻo nā vector kumu i hoʻohana pinepine ʻia ʻo \(\vec{i}\), \(\vec{j}\), a me \(\vec{k}\), e pili ana i nā kuhikuhi x, y, a me z.
4. Nā Hana Vector:
– Hoʻohui: \(\vec{A} + \vec{B}\)
– Ka Hoʻemi ʻana : \(\vec{A} – \vec{B}\)
– Hoʻonui Scalar : \(k\vec{A}\)
– Hoʻonui kiko (huahana kiko): \(\vec{A} \cdot \vec{B}\)
– Hoʻonui Keʻa (huahana keʻa): \(\vec{A} \times \vec{B}\)
Nā ʻAno Vector
Hiki ke loaʻa nā ʻano vectors like ʻole ma muli o ke ʻano a me ko lākou ʻano:
1. Zero Vector: He vector nona ka nui o 0 a ʻaʻohe kuhikuhi. Ua hōʻike ʻia e 0 a i ʻole \(\vec{0}\).
2. Vector Unit: He vector nona ka nui o 1. Hoʻohana pinepine ʻia e hōʻike i ke kuhikuhi.
3. Vector Kūlana: He vector e hōʻike ana i ke kūlana o kahi kiko e pili ana i ke kumu (0,0,0).
4. Nā Vectors Parallel a me Anti-parallel: Nā Vectors e like ana ma ke kuhikuhi a me ke kuhikuhi ʻē aʻe, akā aia ma ka laina hana like.
5. Nā Vectors Coplanar: Nā Vectors ma ka mokulele like.
Nā Nīnau Laʻana a me ke Kūkākūkā
Nīnau 1: Ke helu ʻana i ka nui o ka Vector
He aha ka nui o ka vector \(\vec{A} = (3, 4)\)?
Pane:
No ka helu ʻana i ka nui o ka vector \(\vec{A}\), hoʻohana mākou i ke ʻano hana:
\[\|\vec{A}\| = \sqrt{A_x^2 + A_y^2}\]
E hoʻololi i nā waiwai i loko o ke ʻano:
\[\|\vec{A}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]
No laila, ʻo ka nui o ka vector \(\vec{A}\) he 5.
Nīnau 2: Hoʻohui a me ka Hoʻemi ʻana o nā Vectors
Hāʻawi ʻia ʻelua mau vectors \(\vec{A} = (2, 3)\) a me \(\vec{B} = (1, -1)\). E helu \(\vec{A} + \vec{B}\) a me \(\vec{A} – \vec{B}\).
Pane:
Hoʻohui ʻia o nā vectors \(\vec{A}\) a me \(\vec{B}\):
\[\vec{A} + \vec{B} = (2, 3) + (1, -1) = (2 + 1, 3 – 1) = (3, 2)\]
Ka unuhi ʻana o nā vectors \(\vec{A}\) a me \(\vec{B}\):
\[\vec{A} – \vec{B} = (2, 3) – (1, -1) = (2 – 1, 3 – (-1)) = (1, 4)\]
No laila, \(\vec{A} + \vec{B} = (3, 2)\) a me \(\vec{A} – \vec{B} = (1, 4)\).
Nīnau 3: Huahana Kiko
E helu i ka huahana kiko o nā vectors ʻelua \(\vec{A} = (2, 3)\) a me \(\vec{B} = (1, 4)\).
Pane:
ʻO ka huahana kiko o nā vectors ʻelua penei:
\[\vec{A} \cdot \vec{B} = A_x \cdot B_x + A_y \cdot B_y\]
Hoʻololi waiwai:
\[\vec{A} \cdot \vec{B} = 2 \cdot 1 + 3 \cdot 4 = 2 + 12 = 14\]
No laila, ʻo ka huahana kiko o \(\vec{A}\) a me \(\vec{B}\) he 14.
Nīnau 4: Huahana Kea
Hāʻawi ʻia ʻelua mau vectors i loko o kahi ākea ʻekolu-dimensional \(\vec{A} = (1, 2, 3)\) a me \(\vec{B} = (4, 5, 6)\). E helu i ka huahana kea \(\vec{A} \times \vec{B}\).
Pane:
Ua wehewehe ʻia ka huahana kea o ʻelua mau vectors ma kahi ʻekolu-dimensional ʻo ia ka mea hoʻoholo o ka matrix aʻe:
\[\vec{A} \times \vec{B} =
\begin{vmatrix}
\vec{i} & \vec{j} & \vec{k} \\
ʻA_x & A_y & A_z \\
ʻO B_x & B_y & B_z
\end{vmatrix}
\]
No nā vectors \(\vec{A}\) a me \(\vec{B}\):
\[\vec{A} \times \vec{B} =
\begin{vmatrix}
\vec{i} & \vec{j} & \vec{k} \\
1 & 2 & 3 \\
4 & 5 & 6
\end{vmatrix}
\]
I helu ʻia penei:
\[
\vec{A} \times \vec{B} = \vec{i}(2 \cdot 6 – 3 \cdot 5) – \vec{j}(1 \cdot 6 – 3 \cdot 4) + \vec{k}(1 \cdot 5 – 2 \cdot 4)
\]
\[
= \vec{i}(12 – 15) – \vec{j}(6 – 12) + \vec{k}(5 – 8)
\]
\[
= \vec{i}(-3) – \vec{j}(-6) + \vec{k}(-3)
\]
\[
= -3\vec{i} + 6\vec{j} – 3\vec{k}
\]
No laila, ʻo ka huahana kea o \(\vec{A}\) a me \(\vec{B}\) ʻo \(\vec{A} \times \vec{B} = (-3, 6, -3)\).
I ka wā e hana ai me nā pilikia vector, ʻo ka hoʻomaopopo ʻana i nā manaʻo kumu a me nā huaʻōlelo ke kumu hoʻomaka mua. Manaʻo kēia ʻatikala e hāʻawi i ka poʻe heluhelu i kahi ʻike o nā hana vector like ʻole a me ko lākou mau ʻano like ʻole, kahi mea waiwai nui i ka nānā ʻana i ka makemakika a me ke kino.
