Nā nīnau hoʻohālike a me ke kūkākūkā ʻana o nā ʻāpana Vector
He manaʻo nui nā vectors i ka physics a me ka makemakika, i hoʻohana pinepine ʻia e wehewehe i nā nui me ka nui a me ke kuhikuhi. He mea nui ka hoʻomaopopo piha ʻana i nā vectors no ka hoʻoponopono ʻana i nā pilikia like ʻole i ka ʻepekema a me ka ʻenekinia. E kūkākūkā kēia ʻatikala i kekahi mau pilikia hoʻohālike e pili ana i nā ʻāpana vector, me kā lākou wehewehe ʻana.
Hoʻolauna i nā Vectors
ʻO ka vector kahi nui nona nā ʻano nui ʻelua: ka nui a me ke kuhikuhi. No ka laʻana, ʻo ka wikiwiki he nui vector no ka mea he nui (pehea ka wikiwiki) a me ke kuhikuhi (kahi e hele ai). No ka hōʻike ʻana i nā vectors, hoʻohana pinepine mākou i nā pua, kahi e hōʻike ai ka lōʻihi o ka pua i kona nui a hōʻike ke kuhikuhi o ka pua i kona kuhikuhi.
Hoʻike pinepine ʻia kahi vector i kahi ākea ʻelua-dimensional e like me 𝐀 = 𝑎ᵢ + 𝑏ⱼ, kahi ʻo 𝑎 a me 𝑏 nā ʻāpana o ka vector ma nā ax-x a me y, a ʻo 𝐢 a me 𝐣 nā vectors unit ma nā ax-x a me y.
Laʻana Nīnau 1: Ke Hoʻoholo nei i nā ʻĀpana Vector mai kahi Hōʻike Kiʻi
Nīnau: Loaʻa i kahi vector 𝐀 kahi hoʻomaka ma ke kumu (0,0) a me kahi hopena ma nā hoʻonohonoho (4,3). E hoʻoholo i nā ʻāpana o ka vector 𝐀.
Kūkākūkā: Hiki ke kākau ʻia ke vector e hoʻomaka ana mai ke kiko hoʻomaka (0,0) a i ke kiko hopena (4,3) ma ke ʻano ʻāpana e like me 𝐀 = 4𝐢 + 3𝐣. ʻO ka ʻāpana ma ke axis-x he 4 a ma ke axis-y he 3.
Laʻana Nīnau 2: Ke hoʻoholo nei i ka nui o kahi Vector
Pilikia: E helu i ka nui o ka vector 𝐀 = 4𝐢 + 3𝐣.
Kūkākūkā: Hiki ke helu ʻia ka nui (a i ʻole ka nui) o kahi vector 𝐀 me ka hoʻohana ʻana i ke ʻano Pythagorean, ʻo ia hoʻi:
\[ |𝐀| = \sqrt{𝑎² + 𝑏²} \]
No ke vector 𝐀 = 4𝐢 + 3𝐣, a laila:
\[ |𝐀| = \sqrt{4² + 3²} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
No laila, ʻo ka nui o ka vector 𝐀 he 5 mau ʻāpana.
Laʻana 3: Hoʻohui i ʻelua Vectors
Nīnau: Hāʻawi ʻia ʻelua mau vectors 𝐁 = 2𝐢 + 3𝐣 a me 𝐂 = -𝐢 + 4𝐣. E hoʻoholo i ka huina o nā vectors 𝐁 a me 𝐂.
Kūkākūkā: No ka hoʻohui ʻana i ʻelua vectors, hoʻohui wale mākou i nā ʻāpana ma ke kuhikuhi like o kēlā me kēia vector:
\[ 𝐁 + 𝐂 = (2𝐢 + 3𝐣) + (-𝐢 + 4𝐣) \]
\[ = (2 + (-1))𝐢 + (3 + 4)𝐣 \]
\[ = 1𝐢 + 7𝐣 \]
No laila, ʻo ka hopena o ka hoʻohui ʻana i nā vectors 𝐁 a me 𝐂 ʻo 𝐃 = 𝐢 + 7𝐣.
Laʻana Nīnau 4: Ke helu ʻana i ke kihi ma waena o nā Vectors ʻelua
Pilikia: Hāʻawi ʻia ʻelua mau vectors 𝐀 = 3𝐢 + 4𝐣 a me 𝐁 = 4𝐢 – 3𝐣. E helu i ke kihi ma waena o nā vectors ʻelua.
Kūkākūkā: Hiki ke helu ʻia ke kihi ma waena o nā vectors ʻelua me ka hoʻohana ʻana i ke ʻano cosine:
\[ \cos(𝜃) = \frac{𝐀 · 𝐁}{|𝐀| |𝐁|} \]
1. E helu i ka huahana kiko (𝐀 · 𝐁):
\[ 𝐀 · 𝐁 = (3𝐢 + 4𝐣) · (4𝐢 – 3𝐣) \]
\[ = (3 4) + (4 -3) \]
\[ = 12 – 12 \]
\[ = 0 \]
2. E helu i ka nui o nā vectors 𝐀 a me 𝐁:
\[ |𝐀| = \sqrt{3² + 4²} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
\[ |𝐁| = \sqrt{4² + (-3)²} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
3. E pani i loko o ke ʻano cosine:
\[ \cos(𝜃) = \frac{0}{5 5} = 0 \]
No ka mea, ʻo \(\cos(𝜃) = 0\), a laila \(𝜃 = 90°\). No laila, ʻo ke kihi ma waena o nā vectors ʻelua he 90 degere.
Laʻana Nīnau 5: Ke helu ʻana i ka Hua Hoʻonui Kea o nā Vectors
Pilikia: Hāʻawi ʻia i ʻelua mau vectors i ʻekolu ana, 𝐀 = 𝐢 + 2𝐣 + 3𝐤 a me 𝐁 = 4𝐢 + 5𝐣 + 6𝐤, e helu i ka vector huahana kea 𝐀 × 𝐁.
Kūkākūkā: ʻO ka huahana kea o ʻelua mau vectors ma nā ana ʻekolu (𝐀 × 𝐁) penei:
\[ 𝐀 × 𝐁 = \begin{vmatrix} 𝐢 & 𝐣 & 𝐤 \\ 1 & 2 & 3 \\ 4 & 5 & 6 \end{vmatrix} \]
\[ = 𝐢 (2 6 – 3 5) – 𝐣 (1 6 – 3 4) + 𝐤 (1 5 – 2 4) \]
\[ = 𝐢 (12 – 15) – 𝐣 (6 – 12) + 𝐤 (5 – 8) \]
\[ = 𝐢 (-3) – 𝐣 (-6) + 𝐤 (-3) \]
\[ = -3𝐢 + 6𝐣 – 3𝐤 \]
No laila, ʻo ka hopena o ka huahana kea 𝐀 × 𝐁 he -3𝐢 + 6𝐣 – 3𝐤.
Ka hopena
I ke kinoea a me ka makemakika, he ala pono loa nā vectors e hōʻike i nā nui i loaʻa ke kuhikuhi a me ka nui. Ma ka hoʻomaopopo ʻana pehea e hoʻoholo ai i nā ʻāpana vector, e helu i nā nui, e hoʻohui i nā vectors, a e helu i nā kihi ma waena o nā vectors a me nā huahana kea, hiki iā mākou ke hoʻoponopono i nā pilikia like ʻole e pili ana i nā vectors. ʻO ke kūkākūkā ʻana i nā pilikia hoʻohālike ma luna nei e kōkua i ka hoʻonui ʻana i ko mākou ʻike i kēia manaʻo. ʻO ka mea hope loa, ʻo ka hiki ke hoʻomaopopo a hana me nā vectors he mākaukau pono loa ia ma nā ʻano ʻepekema a me ka ʻenekinia like ʻole.
