He manaʻo koʻikoʻi nā vectors i ka physics, i hoʻohana ʻia e hōʻike i nā nui me ka nui a me ke kuhikuhi. I ka physics, hoʻohana pinepine ʻia nā vectors e wehewehe i nā hanana like ʻole e like me ka ikaika, ka wikiwiki, ka wikiwiki, a me nā mea hou aku. E kūkākūkā kēia ʻatikala i kekahi mau laʻana o nā pilikia vector physics, me kā lākou mau hoʻonā a me nā wehewehe.
1. Hoʻohui a me ka Hoʻemi Vector
Laʻana Nīnau 1:
Ua hāʻawi ʻia ʻelua mau vectors \(\mathbf{A}\) a me \(\mathbf{B}\) penei:
\[
\mathbf{A} = 3\mathbf{i} + 4\mathbf{j}
\]
\[
\mathbf{B} = -2\mathbf{i} + 5\mathbf{j}
\]
E helu:
1. \(\mathbf{A} + \mathbf{B}\)
2. \(\mathbf{A} – \mathbf{B}\)
Hoʻonā:
No ka hoʻohui ʻana i ʻelua vectors, hoʻohui mākou i kā lāua mau ʻāpana ma ke kaʻawale.
1. \(\mathbf{A} + \mathbf{B}\):
\[
\mathbf{A} + \mathbf{B} = (3\mathbf{i} + 4\mathbf{j}) + (-2\mathbf{i} + 5\mathbf{j})
\]
\[
= (3 – 2)\mathbf{i} + (4 + 5)\mathbf{j}
\]
\[
= 1\mathbf{i} + 9\mathbf{j}
\]
\[
\mathbf{A} + \mathbf{B} = \mathbf{i} + 9\mathbf{j}
\]
2. \(\mathbf{A} – \mathbf{B}\):
\[
\mathbf{A} – \mathbf{B} = (3\mathbf{i} + 4\mathbf{j}) – (-2\mathbf{i} + 5\mathbf{j})
\]
\[
= (3 – (-2))\mathbf{i} + (4 – 5)\mathbf{j}
\]
\[
= (3 + 2)\mathbf{i} + (-1)\mathbf{j}
\]
\[
= 5\mathbf{i} – \mathbf{j}
\]
No laila, ʻo ka hopena:
\[
\mathbf{A} – \mathbf{B} = 5\mathbf{i} – \mathbf{j}
\]
2. Hoʻonui Scalar (Huahana Kiko)
Laʻana Nīnau 2:
Ua hāʻawi ʻia ʻelua mau vectors \(\mathbf{C}\) a me \(\mathbf{D}\) penei:
\[
\mathbf{C} = 6\mathbf{i} + 2\mathbf{j}
\]
\[
\mathbf{D} = 3\mathbf{i} + 4\mathbf{j}
\]
E helu i ka huahana scalar (huahana kiko) o \(\mathbf{C}\) a me \(\mathbf{D}\).
Hoʻonā:
ʻO ka huahana scalar o nā vectors ʻelua \(\mathbf{C}\) a me \(\mathbf{D}\) penei:
\[
\mathbf{C} \cdot \mathbf{D} = (6\mathbf{i} + 2\mathbf{j}) \cdot (3\mathbf{i} + 4\mathbf{j})
\]
\[
= 6 \cdot 3 + 2 \cdot 4
\]
\[
= 18 + 8
\]
\[
= 26
\]
No laila, ʻo ka hopena o ka huahana scalar o \(\mathbf{C}\) a me \(\mathbf{D}\) he 26.
3. Huahana Kea
Laʻana Nīnau 3:
Ua hāʻawi ʻia ʻelua mau vectors \(\mathbf{E}\) a me \(\mathbf{F}\) penei:
\[
\mathbf{E} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}
\]
\[
\mathbf{F} = 4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}
\]
E helu i ka huahana kea o \(\mathbf{E}\) a me \(\mathbf{F}\).
Hoʻonā:
Hiki ke helu ʻia ka huahana kea o nā vectors ʻelua \(\mathbf{E}\) a me \(\mathbf{F}\) me ka hoʻohana ʻana i ka matrix determinant:
\[
\mathbf{E} \manawa \mathbf{F} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 & 2 & 3 \\
4 & 5 & 6
\end{vmatrix}
\]
E helu i ka mea hoʻoholo o ka matrix:
\[
\mathbf{E} \times \mathbf{F} = \mathbf{i} (2 \cdot 6 – 3 \cdot 5) – \mathbf{j} (1 \cdot 6 – 3 \cdot 4) + \mathbf{k} (1 \cdot 5 – 2 \cdot 4)
\]
\[
= \mathbf{i} (12 – 15) – \mathbf{j} (6 – 12) + \mathbf{k} (5 – 8)
\]
\[
= \mathbf{i} (-3) – \mathbf{j} (-6) + \mathbf{k} (-3)
\]
\[
= -3\mathbf{i} + 6\mathbf{j} – 3\mathbf{k}
\]
No laila, ʻo ka hopena o ka huahana kea o \(\mathbf{E}\) a me \(\mathbf{F}\) penei:
\[
\mathbf{E} \manawa \mathbf{F} = -3\mathbf{i} + 6\mathbf{j} – 3\mathbf{k}
\]
4. Ka nui o ka Vector
Laʻana Nīnau 4:
Hāʻawi ʻia ke vector \(\mathbf{G} = 3\mathbf{i} – 4\mathbf{j}\). E helu i ka nui (lōʻihi) o ka vector \(\mathbf{G}\).
Hoʻonā:
Hiki ke helu ʻia ka nui o ka vector \(\mathbf{G}\) me ka hoʻohana ʻana i ke ʻano hana:
\[
|\mathbf{G}| = \sqrt{(3)^2 + (-4)^2}
\]
\[
= \sqrt{9 + 16}
\]
\[
= \sqrt{25}
\]
\[
= 5
\]
No laila, ʻo ka nui o ka vector \(\mathbf{G}\) he 5.
5. Hoʻonā Vector
Laʻana Nīnau 5:
ʻO ka nui o ka vector \(\mathbf{H}\) he 10 mau ʻāpana a hana i kahi kihi o 30° me ka axis-x. E hoʻoholo i nā ʻāpana o ka vector \(\mathbf{H}\) ma nā ax-x a me y.
Hoʻonā:
Hiki ke helu ʻia nā ʻāpana o ka vector \(\mathbf{H}\) ma nā axis x (\(\mathbf{H}_x\)) a me y (\(\mathbf{H}_y\)) me ka hoʻohana ʻana i ka trigonometry:
\[
\mathbf{H}_x = |\mathbf{H}| \cos(\theta)
\]
\[
\mathbf{H}_y = |\mathbf{H}| \ hewa(\theta)
\]
Me \(|\mathbf{H}| = 10\) a me \(\theta = 30°\):
\[
H_x = 10 \cos(30°)
\]
\[
H_y = 10 sin(30°)
\]
ʻO nā waiwai o \(\cos(30°) = \frac{\sqrt{3}}{2}\) a me \(\sin(30°) = \frac{1}{2}\):
\[
\mathbf{H}_x = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}
\]
\[
\mathbf{H}_y = 10 \cdot \frac{1}{2} = 5
\]
No laila, ʻo nā ʻāpana o ka vector \(\mathbf{H}\):
\[
\mathbf{H}_x = 5\sqrt{3}
\]
\[
\mathbf{H}_y = 5
\]
Ka hopena
Ma kēia ʻatikala, ua kūkākūkā mākou i kekahi mau pilikia hoʻohālike e pili ana i nā vectors i ka physics, mai ka hoʻohui a me ka hoʻemi ʻana o ka vector, ka hoʻonui scalar a me ke keʻa, a hiki i ka nui a me ka hoʻonā ʻana o ka vector. He mea koʻikoʻi ka hoʻomaopopo ʻana i ke kumumanaʻo a me ka hana ʻana o nā vectors i ka physics no ka mea hiki ke wehewehe ʻia nā hanana kūlohelohe he nui me ka hoʻohana ʻana i nā vectors. Me ka manaʻolana, e kōkua kēia mau pilikia hoʻohālike iā ʻoe e hoʻomaopopo hohonu i ke kumumanaʻo o nā vectors.
