Laʻana o ka Nīnau e Kūkākūkā ana i ka Hoʻohui ʻana o ʻElua Vectors me ka hoʻohana ʻana i ke ʻAno Parallelogram
He manaʻo koʻikoʻi ka hoʻohui vector i ka physics a me ka makemakika, i hoʻohana pinepine ʻia e wehewehe i nā hanana kūlohelohe a me nā pilikia o ke ola o kēlā me kēia lā. Aia kekahi mau ʻano hana no ka hoʻohui ʻana i ʻelua vectors, ʻo kekahi o ia mau mea ʻo ke ʻano parallelogram. ʻAʻole wale kēia ʻano hana intuitive akā hāʻawi pū kekahi i kahi hiʻohiʻona ikaika o ke ʻano o ka hui ʻana o ʻelua vectors e hana i kahi vector hopena. Ma kēia ʻatikala, e nānā mākou i kekahi mau laʻana o ka hoʻohui vector me ka hoʻohana ʻana i ke ʻano parallelogram, me kā lākou mau hoʻonā.
He aha ka Vector?
Ma mua o ko mākou komo ʻana i nā pilikia hoʻohālike, pono mākou e hoʻomaopopo i ka wehewehe kumu o kahi vector. ʻO ka vector kahi nui i loaʻa ka nui (lōʻihi) a me ke kuhikuhi. ʻO nā hiʻohiʻona maʻamau o nā vectors e komo pū me ka wikiwiki, ka wikiwiki, ka ikaika, a me ka neʻe ʻana. Hiki ke hōʻike ʻia kahi vector ma ke ʻano he mau ʻāpana (i, j, k) ma nā hoʻonohonoho Cartesian a i ʻole ma ke ʻano he lōʻihi a me ke kuhikuhi (kihi).
Ke ʻAno Parallelogram
ʻO ke ʻano parallelogram kekahi ala e hoʻohui ai i ʻelua vectors. Ma kēia ʻano hana, hōʻike mākou i ʻelua vectors ma ke ʻano he ʻelua ʻaoʻao o kahi parallelogram. ʻO ka vector hopena ka diagonal o ka parallelogram e hoʻomaka ana mai kahi hoʻomaka o nā vectors ʻelua. Ma ke ʻano makemakika, inā loaʻa iā mākou ʻelua vectors \(\vec{A}\) a me \(\vec{B}\), ʻo ka hopena \( \vec{R} = \vec{A} + \vec{B} \).
ʻO ke ʻano hana maʻamau no ka hoʻohana ʻana i ke ʻano parallelogram penei:
1. E kaha i ka vector \(\vec{A}\) mai ke kiko hoʻomaka.
2. Mai ka hopena o ke vector \(\vec{A}\), e kahakiʻi i ke vector \(\vec{B}\).
3. E kaha i kahi laina e kūlike ana me ka vector \(\vec{B}\) mai ke kiko hoʻomaka \(\vec{A}\).
4. E kaha i kahi laina e kūlike ana me ka vector \(\vec{A}\) mai ka hopena o ka vector \(\vec{B}\).
5. E kaha i kahi diagonal mai ke kiko hoʻomaka a i ke kihi ʻē aʻe e loaʻa ai ka vector hopena \(\vec{R}\).
Nā Nīnau Laʻana a me ke Kūkākūkā
Nīnau 1
Manaʻo mākou he ʻelua mau vectors \(\vec{A}\) a me \(\vec{B}\):
– ʻO ka lōʻihi (nui) o \(\vec{A}\) he 5 mau ʻāpana a me ke kuhikuhi o 0° (a i ʻole ma ke axis-x maikaʻi),
– He 3 anakahi ka lōʻihi o \(\vec{B}\) a he 90° ke kuhikuhi (a i ʻole ma ke axis-y maikaʻi).
He aha ka waiwai hopena o ka hoʻohui ʻana i kēia mau vectors ʻelua me ka hoʻohana ʻana i ke ʻano parallelogram?
Kūkākūkā:
1. E kaha i ka vector \(\vec{A}\) ma ke axis x maikaʻi me ka lōʻihi o 5 mau ʻāpana.
2. Mai ka hopena o ka vector \(\vec{A}\), e kahakiʻi i ka vector \(\vec{B}\) ma ke axis-y maikaʻi me ka lōʻihi o 3 mau ʻāpana.
3. Mai ke kiko hoʻomaka \(\vec{A}\), e kahakiʻi i kahi laina e kūlike ana me \(\vec{B}\).
4. Mai ka hopena o \(\vec{B}\), e kaha i kahi laina e kūlike ana me \(\vec{A}\).
5. ʻO ka hopena he parallelogram me kahi diagonal ʻo ia ka vector hopena \(\vec{R}\).
ʻOiai ʻo \(\vec{A}\) a me \(\vec{B}\) he kū pololei kekahi i kekahi, hiki iā mākou ke hoʻohana i ka Pythagorean theorem e helu i ka lōʻihi o ka vector resultant:
R = \sqrt{A^2 + B^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \approx 5.83 \]
Hiki ke helu ʻia ke kuhikuhi o ka vector resultant me ka hoʻohana ʻana i ka trigonometry. Inā ʻo \(\theta\) ke kihi ma waena o ka resultant a me \(\vec{A}\):
\[ \tan(\theta) = \frac{B}{A} = \frac{3}{5} \]
pēlā:
\[ \theta = \tan^{-1}\left(\frac{3}{5}\right) \approx 30.96^\circ \]
No laila, ʻo ka vector hopena \(\vec{R}\) he nui ma kahi o 5.83 mau ʻāpana a me ke kuhikuhi ma kahi o 30.96° mai \(\vec{A}\).
Nīnau 2
Ua hāʻawi ʻia ʻelua mau vectors \(\vec{C}\) a me \(\vec{D}\) penei:
– \(\vec{C}\) me ka lōʻihi o 4 mau anakahi a me ke kuhikuhi o 45°.
– \(\vec{D}\) me ka lōʻihi o 6 mau anakahi a me ke kuhikuhi o 120°.
E hoʻoholo i ka vector hopena \(\vec{R}\) mai ka hoʻohui ʻana o nā vectors ʻelua.
Kūkākūkā:
No ka hoʻohui ʻana i ʻelua mau vectors i kū ʻole kekahi i kekahi a i ʻole ma nā ʻano like ʻole, hiki iā ʻoe ke hoʻohana i nā ʻāpana Cartesian.
1. E wāwahi iā \(\vec{C}\) a me \(\vec{D}\) i nā ʻāpana x a me y.
No \(\vec{C}\):
\[ C_x = C \cos(45^\circ) = 4 \cos(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2} \approx 2.83 \]
\[ C_y = C \sin(45^\circ) = 4 \sin(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2} \approx 2.83 \]
No \(\vec{D}\):
\[ D_x = D \cos(120^\circ) = 6 \cos(120^\circ) = 6 \cdot (-\frac{1}{2}) = -3 \]
\[ D_y = D \sin(120^\circ) = 6 \sin(120^\circ) = 6 \cdot \frac{\sqrt{3}}{2} = 3\sqrt{3} \approx 5.20 \]
2. E hoʻohui i nā ʻāpana x a me y o nā vectors ʻelua:
\[ R_x = C_x + D_x = 2.83 + (-3) = -0.17 \]
\[ R_y = C_y + D_y = 2.83 + 5.20 = 8.03 \]
3. E helu i ka nui a me ke kuhikuhi o ka vector hopena \(\vec{R}\):
R = \sqrt{R_x^2 + R_y^2} = \sqrt{(-0.17)^2 + 8.03^2} = \sqrt{0.03 + 64.48} = \sqrt{64.51} \approx 8.03 \]
\[ \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{8.03}{-0.17}\right) \approx \tan^{-1}(-47.24) \]
ʻOiai he maikaʻi ʻole ka hopena, hoʻohui mākou i 180° e loaʻa ai ke kihi ma ka ʻōnaehana quadrant kūpono:
\[ \theta \approx \tan^{-1}(47.24) + 180^\circ \approx 271.93^\circ \]
No laila, ʻo ka vector hopena \(\vec{R}\) he nui ma kahi o 8.03 mau ʻāpana a me ke kuhikuhi ma kahi o 271.93°, a i ʻole hiki iā mākou ke ʻōlelo ma kahi o 91.93° mai ka axis-x maikaʻi ʻole ma ka quadrant ʻehā.
Pani
He ala kūpono a ʻike maka ke ʻano parallelogram e hoʻohui i ʻelua vectors. ʻOiai ke ʻano maʻalahi kēia ʻano no nā vectors maʻalahi, he mea nui e hoʻomaopopo no nā vectors paʻakikī, pono pinepine mākou e hoʻohana i nā ʻāpana Cartesian a me nā ʻano hana algebraic holomua e loaʻa ai nā hopena pololei. Me ka manaʻolana, hāʻawi nā laʻana ma luna i kahi kiʻi maopopo o ke ʻano e hiki ai ke hoʻopili ʻia kēia ʻano ma nā kūlana like ʻole.
