ʻO ke ʻano Vector Resultant: Manaʻo, ʻAno Hana, a me nā pilikia hoʻohālike
ʻO ka vector kahi nui i loaʻa ka nui a me ke kuhikuhi. I ka physics a me ka makemakika, hoʻohana pinepine ʻia nā vectors e wehewehe i nā hanana like ʻole e like me ka wikiwiki, ka ikaika, a me ka neʻe ʻana. ʻO ka helu ʻana i ka vector hopena, ka huina o ʻelua a ʻoi aku paha nā vectors, he mākaukau koʻikoʻi ia i hoʻohana pinepine ʻia i nā noi ʻepekema a me nā ʻenehana like ʻole. E kūkākūkā kēia ʻatikala i ke kumumanaʻo kumu o nā vectors, nā ʻano hana no ka helu ʻana i ka vector hopena, a hāʻawi i kekahi mau pilikia hoʻohālike e hoʻomālamalama i ka hoʻomaopopo ʻana.
Ke hoʻomaopopo nei i nā Vectors a me nā Vectors Resultant
ʻalalā
He mea makemakika ka vector nona nā ʻano nui ʻelua:
1. Ka nui: Ka nui o ka waiwai vector.
2. Kuhikuhi: Hōʻike ke kuhikuhi o kahi vector i ke kuhikuhi o ka vector ma ka lewa.
Hoʻike pinepine ʻia nā vectors ma ke ʻano he mau pua, kahi e hōʻike ai ka lōʻihi o ka pua i ka nui a ʻo ke kuhikuhi o ka pua e hōʻike ana i ke kuhikuhi o ka vector.
Vector hopena
ʻO kahi vector resultant he vector hoʻokahi e hōʻike ana i ka hui pū ʻana o ʻelua a ʻoi aku paha nā vectors. ʻO ke kaʻina hana o ka hoʻohui ʻana i nā vectors ua kapa ʻia hoʻi ʻo "vector addition." Aia kekahi mau ʻano hana e hiki ke hoʻohana ʻia e helu i ka vector resultant, me nā ʻano hana kiʻi a me nā ʻano loiloi.
ʻAno Helu Hua Vector
ʻAno Hana Kiʻi
ʻO ke ʻano hana kiʻi e pili ana i ka hōʻike ʻana i nā vectors ma ke ʻano geometric a me ka hoʻohana ʻana i nā lula o ka hoʻohui vector e ʻike i ka hopena. ʻO nā lula nui ʻelua o ke ʻano hana kiʻi:
1. Ke ʻAno Huinakolu: Ma kēia ʻano hana, ua kaha ʻia ka vector ʻelua mai ka hopena o ka vector mua. ʻO ka vector hopena ka vector i kaha ʻia mai ka hoʻomaka ʻana o ka vector mua a i ka hopena o ka vector ʻelua.
2. Ke ʻAno Polygon: Hoʻohana ʻia kēia ʻano hana e hoʻohui i nā vectors ʻoi aku ma mua o ʻelua. Ua kahakiʻi ʻia nā vectors ma ke ʻano kaʻina mai ka hopena a i ka hopena, a ʻo ka vector hopena ʻo ia ka vector e hoʻopili ana i ke kiko hoʻomaka o ka vector mua i ka hopena o ka vector hope loa.
Ke ʻAno Hana Loiloi
ʻO ke ʻano hana loiloi e pili ana i ka hoʻohana ʻana i ka makemakika a me ka trigonometry e helu i ka vector hopena. ʻO nā ʻano hana nui ʻelua i ke ʻano hana loiloi:
1. ʻAno Hana ʻĀpana: Ma kēia ʻano hana, ua hoʻokaʻawale ʻia kēlā me kēia vector i loko o kona mau ʻāpana ma nā axes x- a me y. A laila hoʻohui ʻia kēia mau ʻāpana e loaʻa ai nā ʻāpana o ka vector hopena. ʻO ka hope loa, ua helu ʻia ka vector hopena me ka hoʻohana ʻana i ka theorem Pythagorean a me ka trigonometry.
2. Ke ʻAno Cosine: Hoʻohana ʻia kēia ʻano hana ke ʻike ʻia nā nui o nā vectors ʻelua a me ke kihi ma waena o lākou. Hoʻohana ʻia ke ʻano cosine e helu i ka nui o ka vector hopena.
Nā Haʻilula Vector Hualoaʻa
ʻAno Hana ʻĀpana
No nā vectors ʻelua \(\mathbf{A}\) a me \(\mathbf{B}\) me nā ʻāpana:
\[
\mathbf{A} = A_x \hat{i} + A_y \hat{j}
\]
\[
\mathbf{B} = B_x \hat{i} + B_y \hat{j}
\]
ʻO ke vector hopena \(\mathbf{R}\) penei:
\[
\mathbf{R} = \mathbf{A} + \mathbf{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j}
\]
Hiki ke helu ʻia ka nui o ka vector hopena \(\mathbf{R}\) me ka hoʻohana ʻana i ka Pythagorean theorem:
\[
|\mathbf{R}| = \sqrt{(A_x + B_x)^2 + (A_y + B_y)^2}
\]
Ua hoʻoholo ʻia ke kuhikuhi o ka vector hopena e ke kihi \(\theta\) i hoʻokumu ʻia me ka axis-x:
\[
\theta = \tan^{-1}\left(\frac{A_y + B_y}{A_x + B_x}\ʻākau)
\]
Ke ʻAno Cosine
Inā he mau nui ko ʻelua mau vectors \(\mathbf{A}\) a me \(\mathbf{B}\) \(A\) a me \(B\) a he kihi \(\theta\) ma waena o lāua, ʻo ka nui o ka vector hopena \(\mathbf{R}\) penei:
\[
|\mathbf{R}| = \sqrt{A^2 + B^2 + 2AB \cos \theta}
\]
Hiki ke helu ʻia ke kuhikuhi o ka vector hopena me ka hoʻohana ʻana i ke ʻano trigonometric:
\[
ʻO ka tan alpha = B sin theta {A + B cos theta}
\]
Ma kahi o \(\alpha\) ke kihi i hoʻokumu ʻia e ka vector hopena me ka vector \(\mathbf{A}\).
Laʻana o ka pilikia Vector Resultant
Laʻana Nīnau 1: ʻAno Hana ʻĀpana
Nīnau:
ʻElua mau vectors \(\mathbf{A}\) a me \(\mathbf{B}\) i loaʻa nā ʻāpana penei:
\[
\mathbf{A} = 3\papale {i} + 4\papale{j}
\]
\[
\mathbf{B} = 1\papale{i} + 2\papale{j}
\]
E helu i ka vector hopena \(\mathbf{R}\).
Hoʻonā:
1. E hoʻohui i nā ʻāpana ma nā koʻi x a me y:
\[
R_x = A_x + B_x = 3 + 1 = 4
\]
\[
R_y = A_y + B_y = 4 + 2 = 6
\]
2. E helu i ka nui o ka vector hopena:
\[
|\mathbf{R}| = \sqrt{R_x^2 + R_y^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 7,21
\]
3. E helu i ke kuhikuhi o ka vector hopena:
\[
\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{6}{4}\right) = \tan^{-1}(1,5) = 56,31^\circ
\]
No laila, ʻo ka nui o ka vector hopena \(\mathbf{R}\) he 7,21 a me ke kuhikuhi o 56,31 degere i ka axis-x.
Laʻana Nīnau 2: Ke ʻAno Cosine
Nīnau:
ʻElua mau vectors \(\mathbf{A}\) a me \(\mathbf{B}\) he mau anakahi nui \(A = 5\), \(B = 7\) mau anakahi, a ʻo ke kihi ma waena o lāua he 60°. E helu i ka nui o ka vector hopena \(\mathbf{R}\).
Hoʻonā:
1. E hoʻohana i ke ʻano cosine e helu i ka nui o ka vector hopena:
\[
|\mathbf{R}| = \sqrt{A^2 + B^2 + 2AB \cos \theta}
\]
\[
|\mathbf{R}| = \sqrt{5^2 + 7^2 + 2 \cdot 5 \cdot 7 \cdot \cos 60^\circ}
\]
\[
|\mathbf{R}| = \sqrt{25 + 49 + 70 \cdot 0,5}
\]
\[
|\mathbf{R}| = \sqrt{25 + 49 + 35}
\]
\[
|\mathbf{R}| = \sqrt{109} = 10,44 \, \text{unit}
\]
No laila, ʻo ka nui o ka vector hopena \(\mathbf{R}\) he 10,44 mau ʻāpana.
Laʻana 3: Ka hopena o nā Vectors ʻEkolu
Nīnau:
ʻEkolu mau vectors \(\mathbf{A}\), \(\mathbf{B}\), a me \(\mathbf{C}\) me kēia mau ʻāpana:
\[
\mathbf{A} = 2\papale {i} + 3\papale{j}
\]
\[
\mathbf{B} = -1\pale {i} + 4\papale{j}
\]
\[
\mathbf{C} = 3\papale{i} – 2\papale{j}
\]
E helu i ka vector hopena \(\mathbf{R}\).
Hoʻonā:
1. E hoʻohui i nā ʻāpana ma nā koʻi x a me y:
\[
R_x = A_x + B_x + C_x = 2 – 1 + 3 = 4
\]
\[
R_y = A_y + B_y + C_y = 3 + 4 – 2 = 5
\]
2. E helu i ka nui o ka vector hopena:
\[
|\mathbf{R}| = \sqrt{R_x^2 + R_y^2} = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} = 6,4
\]
3. E helu i ke kuhikuhi o ka vector hopena:
\[
\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{5}{4}\right) = \tan^{-1}(1,25) = 51,34^\circ
\]
No laila, ʻo ka vector hopena \(\mathbf{
He 6,4 ka nui o R}\) a he 51,34 kekelē ke kuhikuhi ʻana i ka axis-x.
Ka hopena
ʻO ka helu ʻana i ka hopena o kahi vector he mākaukau koʻikoʻi ia i ka physics a me ka makemakika. Ma ka hoʻohana ʻana i nā ʻano hana kiʻi a i ʻole analytical, hiki iā mākou ke hoʻoholo i ka hopena o ʻelua a ʻoi aku paha nā vectors. ʻO ke ʻano o nā ʻāpana a me ke ʻano o nā cosine ʻelua mau ʻenehana koʻikoʻi i nā helu analytical e hiki ai iā mākou ke helu pololei i ka nui a me ke kuhikuhi o ka vector resultant. Hōʻike nā hiʻohiʻona ma luna i ka hoʻohana pono ʻana o kēia mau manaʻo, e kōkua ana iā mākou e hoʻomaopopo a hoʻohana i nā vectors i nā ʻano ʻepekema a me nā ʻenehana like ʻole.
