Nā nīnau hoʻohālike a me ke kūkākūkā ʻana no ka hoʻohui ʻana i ʻelua mau vectors me ka hoʻohana ʻana i ke ʻano o ka triangle
Pendahuluan
ʻO ka vector kahi nui i loaʻa ka nui a me ke kuhikuhi. I ka physics a me ka makemakika, ʻo ka hoʻomaopopo ʻana pehea e hoʻohui ai i ʻelua vectors he mea nui ia no ka hoʻoponopono ʻana i nā pilikia like ʻole. Aia kekahi mau ʻano hana no ka hoʻohui ʻana i nā vectors, ʻo kekahi o ia mau mea ʻo ke ʻano triangle. Ma kēia ʻatikala, e kūkākūkā mākou i nā laʻana a kūkākūkā i ka hoʻohui ʻana i ʻelua vectors me ka hoʻohana ʻana i ke ʻano triangle me nā kikoʻī.
ʻAno Huinakolu ma ka Hoʻohui Vector
Ma mua o ko kākou komo ʻana i ka pilikia hoʻohālike, e hoʻomaopopo mua kākou pehea e hoʻohana ʻia ai ke ʻano o ka triangle e hoʻohui i ʻelua vectors. ʻO ke ʻano o ka triangle e pili ana i kēia mau ʻanuʻu:
1. Ke kau ʻana i ʻelua mau vectors ma kahi kiko like: Ua kau ʻia ka vector mua i kahi e waiho ai kona huelo (kiko hoʻomaka) ma kahi hoʻomaka i koho ʻia.
2. Ke wehewehe nei i ka Vector ʻElua: Hoʻohui ʻia ka vector ʻelua i ka hopena (kiko hopena) o ka vector mua.
3. Ke hoʻoholo nei i ka Vector Hopena: ʻO ka vector hopena ka vector e hoʻopili ana i ke kiko hoʻomaka o ka vector mua i ke kiko hopena o ka vector ʻelua.
Ka Hoʻopaʻa ʻana o Vector
No nā kumu o kēia ʻatikala, e hoʻohana mākou i ka hōʻailona vector penei:
– Nā Vectors i kākau ʻia me ka wiwo ʻole a i ʻole me ka pua ma luna (no ka laʻana, A a i ʻole \(\vec{A}\)).
– Ua kākau ʻia nā ʻāpana vector ma nā kuhikuhi \(x\) a me \(y\) ma ke ʻano \(A_x\) a me \(A_y\) no ka vector \(\vec{A}\).
Laʻana pilikia
I kēia manawa, e nānā kākou i kahi pilikia hoʻohālike e kōkua iā mākou e hoʻomaopopo i ka hoʻohui ʻana o ʻelua mau vectors me ka hoʻohana ʻana i ke ʻano triangle.
Nīnau:
Hāʻawi ʻia ʻelua mau vectors A a me B penei:
– ʻO ka nui o ka Vector A he 4 mau anakahi a me ke kuhikuhi he 30 degere i ka hikina ʻākau.
– ʻO ka nui o ka Vector B he 3 mau anakahi a me ke kuhikuhi he 60 degere i ka hikina ʻākau.
E hoʻoholo i ka vector hopena R mai ka hoʻohui ʻana o nā vectors ʻelua me ka hoʻohana ʻana i ke ʻano huinakolu.
Pahana
KaʻAnuʻu Hana 1: Ke Kaha Kiʻi ʻana i nā Vectors
ʻO ka mea mua, kahakiʻi mākou i ka vector A me ka nui o 4 mau ʻāpana a me ke kuhikuhi o 30 degere i ka hikina ʻākau. A laila, mai ka hopena o ka vector A, kahakiʻi mākou i ka vector B me ka nui o 3 mau ʻāpana a me ke kuhikuhi o 60 degere i ka hikina ʻākau.
KaʻAnuʻu Hana 2: Ke Helu ʻana i nā ʻĀpana Vector
A laila, helu mākou i nā ʻāpana o kēlā me kēia vector ma nā kuhikuhi \(x\) a me \(y\).
Nā ʻāpana o ka vector \(\vec{A}\):
\[
ʻA_x = A \cos \theta_1 = 4 \cos 30^\circ = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3}
\]
\[
ʻA_y = A \sin \theta_1 = 4 \sin 30^\circ = 4 \times \frac{1}{2} = 2
\]
Nā ʻāpana o ka vector \(\vec{B}\):
\[
B_x = B \cos \theta_2 = 3 \cos 60^\circ = 3 \times \frac{1}{2} = 1.5
\]
\[
B_y = B \sin \theta_2 = 3 \sin 60^\circ = 3 \times \frac{\sqrt{3}}{2} = 1.5\sqrt{3}
\]
KaʻAnuʻu Hana 3: Hoʻohui i nā ʻĀpana Vector
Hoʻohui mākou i nā ʻāpana o nā vectors ʻelua e loaʻa ai nā ʻāpana o ka vector hopena \(\vec{R}\).
\[
R_x = A_x + B_x = 2\sqrt{3} + 1.5
\]
\[
R_y = A_y + B_y = 2 + 1.5\sqrt{3}
\]
KaʻAnuʻu Hana 4: E helu i ka nui a me ke kuhikuhi o ka Vector Resultant
Ua helu ʻia ka nui o ka vector hopena \(\vec{R}\) me ka hoʻohana ʻana i ka Pythagorean theorem:
\[
R = \sqrt{R_x^2 + R_y^2}
\]
\[
R_x = 2\sqrt{3} + 1.5 \approx 3.464 + 1.5 = 4.964
\]
\[
R_y = 2 + 1.5\sqrt{3} \approx 2 + 2.598 = 4.598
\]
\[
R = \sqrt{(4.964)^2 + (4.598)^2} \approx \sqrt{24.640 + 21.145} \approx \sqrt{45.785} \approx 6.75 \text{ mau ʻāpana}
\]
Ua helu ʻia ke kuhikuhi o ka vector hopena \(\vec{R}\) me ka hoʻohana ʻana i ka hana trigonometric tangent:
\[
\tan \phi = \frac{R_y}{R_x} = \frac{4.598}{4.964} \approx 0.926
\]
\[
\phi = \tan^{-1}(0.926) \approx 42.6^\circ \text{ mai ka ʻākau hikina}
\]
Ka hopena
Mai nā hopena ma luna, hiki iā mākou ke hoʻoholo he 6.75 mau ʻāpana ka nui o ka vector hopena \(\vec{R}\) mai ka hoʻohui ʻana o nā vectors \(\vec{A}\) a me \(\vec{B}\) me ka hoʻohana ʻana i ke ʻano triangle a he 42.6 degere mai ka hikina ʻākau mai.
Pani
ʻO ka hoʻohui ʻana i ʻelua mau vectors me ka hoʻohana ʻana i ke ʻano triangle he ʻano hana pono loa ia i hoʻohana pinepine ʻia i ka physics a me ka ʻenekinia. Ma ke kaha kiʻi ʻana i nā vectors a me ka hoʻohui ʻana i kā lākou mau ʻāpana, hiki iā mākou ke loaʻa maʻalahi i ka vector resultant. Manaʻolana, ua kōkua kēia ʻatikala iā ʻoe e hoʻomaopopo i ke kumumanaʻo o ka hoʻohui vector me ka hoʻohana ʻana i ke ʻano triangle a hiki ke hoʻopili ʻia i nā pilikia like ʻole āu e hālāwai ai i kāu mau haʻawina.
