Nā nīnau hoʻohālike e kūkākūkā ana i nā Vectors a me kā lākou hana
He manaʻo nui nā vectors i ka makemakika a me ke kino, hoʻohana pinepine ʻia ma nā ʻano ʻepekema like ʻole. Hōʻike nā vectors i nā nui me ka nui a me ke kuhikuhi. Aia ma lalo iho kekahi mau pilikia hoʻohālike e pili ana i nā vectors a me nā kūkākūkā ʻana o nā hana like ʻole e like me ka hoʻohui, ka hoʻemi ʻana, a me ka hoʻonui ʻana e nā scalars. E hāʻawi kēia ʻatikala i kahi ʻike hohonu o ke ʻano o ka hoʻoponopono ʻana i nā pilikia e pili ana i nā vectors.
1. Hoʻohui Vector
Laʻana Nīnau 1
Hāʻawi ʻia ʻelua mau vectors ma ke ʻano ʻāpana:
ʻA = (3, 4)
B = (1, 2)
E helu i ka hopena o ka hoʻohui ʻana i nā vectors A a me B.
Pahana
Hana ʻia ka hoʻohui vector ma ka hoʻohui ʻana i nā ʻāpana like o nā vectors ʻelua. No laila, hiki iā mākou ke helu
\[
A + B = (3 + 1, 4 + 2) = (4, 6)
\]
No laila, ʻo ka hopena o ka hoʻohui ʻana i nā vectors A a me B ʻo (4, 6).
2. Ka Hoʻemi Vector
Laʻana Nīnau 2
Hāʻawi ʻia ʻelua mau vectors ma ke ʻano ʻāpana:
C = (5, 7)
D = (2, 3)
E helu i ka hopena o ka unuhi ʻana i ka vector C mai ka vector D.
Pahana
Hana ʻia ka unuhi vector ma ka unuhi ʻana i nā ʻāpana like o nā vectors ʻelua. No laila, hiki iā mākou ke helu
\[
C – D = (5 – 2, 7 – 3) = (3, 4)
\]
No laila, ʻo ka hopena o ka unuhi ʻana i nā vectors C a me D ʻo (3, 4).
3. Hoʻonui ʻia o nā Vectors e nā Scalars
Laʻana Nīnau 3
Hāʻawi ʻia ka vector E = (4, -2) a me ka scalar k = 3. E helu i ka hopena o ka hoʻonui ʻana i ka vector E me ka scalar k.
Pahana
Hana ʻia ka hoʻonui ʻana o kahi vector e kahi scalar ma ka hoʻonui ʻana i kēlā me kēia ʻāpana o ka vector e ka scalar. No laila, hiki iā mākou ke helu
\[
k E = 3 (4, -2) = (3 4, 3 -2) = (12, -6)
\]
No laila, ʻo ka hopena o ka hoʻonui ʻana i ka vector E e ka scalar k ʻo ia ka (12, -6).
4. Huahana Kiko
Laʻana Nīnau 4
Hāʻawi ʻia ʻelua mau vectors ma ke ʻano ʻāpana:
F = (1, 3)
G = (4, 2)
E helu i ka huahana kiko o nā vectors F a me G.
Pahana
ʻO ka huahana kiko o nā vectors ʻelua ka huina o nā huahana o kā lākou mau ʻāpana pili. No laila, hiki iā mākou ke helu
\[
F \cdot G = (1 4) + (3 2) = 4 + 6 = 10
\]
No laila, ʻo ka huahana kiko o nā vectors F a me G he 10.
5. Huahana Kea
Laʻana Nīnau 5
Hāʻawi ʻia i ʻelua mau vectors ma 3D:
H = (2, -3, 1)
ʻO wau = (1, 4, -2)
E helu i ka huahana kea o nā vectors H a me I.
Pahana
Hoʻopuka ʻia ka huahana kea o ʻelua mau vectors 3-dimensional e ka mea hoʻoholo o ka matrix e loaʻa ana nā ʻāpana o nā vectors ʻelua. Loaʻa i ka vector hopena nā ʻāpana penei:
\[
H \times I = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
2 & -3 & 1 \\
1 & 4 & -2 \\
\end{vmatrix}
\]
Ma ka helu ʻana i ka determinant, loaʻa iā mākou:
\[
H manawa I = (\mathbf{i}((-3)(-2) – (1)(4)) – \mathbf{j}(2(-2) – (1)(1)) + \mathbf{k}(2(4) – (-3)(1)))
\]
\[
= (\mathbf{i}(6 – 4) – \mathbf{j}(-4 – 1) + \mathbf{k}(8 + 3))
\]
\[
= (\mathbf{i}(2) – \mathbf{j}(-5) + \mathbf{k}(11))
\]
\[
= (2, 5, 11)
\]
No laila, ʻo ka huahana kea o nā vectors H a me I ʻo (2, 5, 11).
6. Ka lōʻihi a i ʻole ka nui o ka Vector
Laʻana Nīnau 6
Hāʻawi ʻia kahi vector J = (6, 8). E helu i ka lōʻihi (ka nui) o ka vector J.
Pahana
Ua helu ʻia ka lōʻihi (ka nui) o kahi vector me ka hoʻohana ʻana i ke ʻano hana:
\[
\| J \| = \sqrt{x^2 + y^2}
\]
Ma kēia hihia, \( x = 6 \) a me \( y = 8 \), no laila:
\[
\| J \| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]
No laila, ʻo ka lōʻihi (ka nui) o ka vector J he 10.
7. Vector Unit
Laʻana Nīnau 7
Hāʻawi ʻia kahi vector K = (-5, 12). E huli i ka vector unit o K.
Pahana
ʻO ka vector unit he vector nona ka lōʻihi o 1. No ka loaʻa ʻana o ka vector unit o kahi vector, pono mākou e puʻunaue i kēlā me kēia ʻāpana o ka vector e ka lōʻihi (ka nui) o ka vector. Hiki ke helu ʻia ka lōʻihi o ka vector K penei:
\[
\| K \|= \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]
A laila ʻo ke vector unit K:
\[
\hat{K} = \left(\frac{-5}{13}, \frac{12}{13}\right)
\]
No laila, ʻo ke vector unit o ka vector K ʻo \(\left(\frac{-5}{13}, \frac{12}{13}\right)\).
Ka hopena
Ma o nā hiʻohiʻona ma luna, ua ʻike mākou pehea e hana ai nā vectors a me kā lākou hana ma nā ʻano like ʻole. ʻO ka hoʻohui a me ka unuhi ʻana o Vector e pili ana i ka hoʻohui a me ka unuhi ʻana i nā ʻāpana like. Hiki ke hana ʻia ka hoʻonui ʻana o Vector ma ke ʻano huahana scalar a i ʻole dot, a ma ke ʻano huahana cross no nā vectors 3D. Hiki iā mākou ke hoʻoholo i ka lōʻihi o kahi vector a loaʻa i kāna vector unit.
He mea koʻikoʻi ka hoʻomaopopo ʻana i kēia mau manaʻo kumu no ka mea hoʻohana ʻia nā vectors i nā noi he nui ma nā ʻano kahua like ʻole, me ke kino, ka ʻenekinia, a me nā kiʻi kamepiula. Me ka lawa ʻana o ka hoʻomaʻamaʻa, hiki iā mākou ke hoʻokele i kēia mau hana a hoʻopili iā lākou i ka nānā ʻana a me ka hoʻoponopono pilikia paʻakikī.
