Laʻana o nā nīnau kūkākūkā hana Vector
ʻO nā hana vector kahi manaʻo nui i ka makemakika e ʻike pinepine ʻia ma nā ʻano noiʻi like ʻole, e like me ke kino, ka ʻenekinia, a me ka ʻepekema kamepiula. Ma kēia ʻatikala, e kūkākūkā mākou i kekahi mau laʻana o nā hana vector a me kā lākou mau hoʻonā e hāʻawi i kahi ʻike hohonu a paʻa. E uhi kēia mau laʻana i nā hana kumu e like me ka hoʻohui a me ka hoʻemi vector, a me nā hana holomua e like me ka hoʻonui scalar a me ka hoʻonui cross-vector.
1. Hoʻohui a me ka Hoʻemi Vector
Laʻana Nīnau 1
Hāʻawi ʻia ʻelua mau vectors A a me B ma ke ʻano ʻāpana:
\[ \mathbf{A} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} \]
\[ \mathbf{B} = \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} \]
E helu i ka hopena o ka hoʻohui a me ka unuhi ʻana o nā vectors ʻelua.
Pahana
No ka hoʻohui vector, hoʻohui mākou i kēlā me kēia ʻāpana pili o nā vectors ʻelua.
\[ \mathbf{A} + \mathbf{B} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} + \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 + (-1) \\ 3 + 4 \\ -1 + 2 \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \\ 1 \end{pmatrix} \]
No ka hoʻemi ʻana i ka vector, hoʻemi mākou i kēlā me kēia ʻāpana pili o nā vectors ʻelua.
\[ \mathbf{A} – \mathbf{B} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} – \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 – (-1) \\ 3 – 4 \\ -1 – 2 \end{pmatrix} = \begin{pmatrix} 3 \\ -1 \\ -3 \end{pmatrix} \]
2. Hoʻonui Scalar ma o ka Vector
Laʻana Nīnau 2
Hāʻawi ʻia kahi vector C a me kahi scalar k:
\[ \mathbf{C} = \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix} \]
\[ k = 4 \]
E helu i ka huahana scalar o ka vector C e ka scalar k.
Pahana
Hana ʻia ka hoʻonui ʻana o kahi scalar me kahi vector ma ka hoʻonui ʻana i kēlā me kēia ʻāpana o ka vector me ka scalar.
\[ k \mathbf{C} = 4 \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix} = \begin{pmatrix} 4 \cdot 1 \\ 4 \cdot (-2) \\ 4 \cdot 3 \end{pmatrix} = \begin{pmatrix} 4 \\ -8 \\ 12 \end{pmatrix} \]
3. Huahana Kiko
Laʻana Nīnau 3
Hāʻawi ʻia ʻelua mau vectors D a me E:
\[ \mathbf{D} = \begin{pmatrix} 3 \\ -2 \\ 4 \end{pmatrix} \]
\[ \mathbf{E} = \hoomaka{pmatrix} 1 \\ 0 \\ -1 \hope{pmatrix} \]
E helu i ka huahana kiko o nā vectors ʻelua.
Pahana
Loaʻa ka huahana kiko o nā vectors ʻelua ma ka hoʻohui ʻana i nā huahana o kā lāua mau ʻāpana pili.
\[ \mathbf{D} \cdot \mathbf{E} = 3 \cdot 1 + (-2) \cdot 0 + 4 \cdot (-1) = 3 + 0 – 4 = -1 \]
4. Huahana Kea
Laʻana Nīnau 4
Hāʻawi ʻia i ʻelua mau vectors F a me G:
\[ \mathbf{F} = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} \]
\[ \mathbf{G} = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix} \]
E helu i ka huahana kea o nā vectors ʻelua.
Pahana
Loaʻa ka huahana kea o ʻelua mau vectors ma kahi ʻekolu-dimensional ma ka hoʻohana ʻana i ka determinant o ka matrix i hoʻokumu ʻia e kēlā mau vectors. Hāʻawi ʻia ka huahana kea e ke ʻano:
\[ \mathbf{F} \manawa \mathbf{G} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 3 & 4 \\ 1 & -1 & 2 \end{vmatrix} \]
Hiki ke helu ʻia kēia ma ke ʻano penei:
\[
\mathbf{F} \manawa \mathbf{G} = \mathbf{i} \begin{vmatrix} 3 & 4 \\ -1 & 2 \end{vmatrix} – \mathbf{j} \begin{vmatrix} 2 & 4 \\ 1 & 2 \end{vmatrix} + \v} 3 & -1 \hope{vmatrix}
\]
Ke helu nei i ka mea hoʻoholo o kēlā me kēia submatrix:
\[
= \mathbf{i} (3 \cdot 2 – 4 \cdot -1) – \mathbf{j} (2 \cdot 2 – 4 \cdot 1) + \mathbf{k} (2 \cdot -1 – 3 \cdot 1)
\]
\[
= \mathbf{i} (6 + 4) – \mathbf{j} (4 – 4) + \mathbf{k} (-2 – 3)
\]
\[
= \mathbf{i} (10) – \mathbf{j} (0) + \mathbf{k} (-5)
\]
\[
= \begin{pmatrix} 10 \\ 0 \\ -5 \end{pmatrix}
\]
No laila, ʻo ka huahana kea o F a me G penei:
\[ \mathbf{F} \times \mathbf{G} = \begin{pmatrix} 10 \\ 0 \\ -5 \end{pmatrix} \]
5. Ke hoʻoholo nei i ke kihi ma waena o nā Vectors ʻelua
Laʻana Nīnau 5
Hāʻawi ʻia ʻelua mau vectors H a me I:
\[ \mathbf{H} = \begin{pmatrix} 6 \\ 2 \\ 3 \end{pmatrix} \]
\[ \mathbf{I} = \begin{pmatrix} 1 \\ 4 \\ -2 \end{pmatrix} \]
E hoʻoholo i ke kihi ma waena o nā vectors ʻelua.
Pahana
Hiki ke loaʻa ke kihi \(\theta\) ma waena o nā vectors ʻelua ma ka hoʻohana ʻana i ka pilina ma waena o ka huahana kiko a me nā nui o nā vectors ʻelua:
\[ \mathbf{H} \cdot \mathbf{I} = \| \mathbf{H} \| \| \mathbf{I} \| \cos \theta \]
ʻO ka mea mua, e helu i ka huahana kiko \( \mathbf{H} \cdot \mathbf{I} \):
\[ \mathbf{H} \cdot \mathbf{I} = 6 \cdot 1 + 2 \cdot 4 + 3 \cdot (-2) = 6 + 8 – 6 = 8 \]
A laila, e helu i ka nui o nā vectors ʻelua:
\[ \| \mathbf{H} \| = \sqrt{6^2 + 2^2 + 3^2} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]
\[ \| \mathbf{I} \| = \sqrt{1^2 + 4^2 + (-2)^2} = \sqrt{1 + 16 + 4} = \sqrt{21} \]
A laila, e hoʻololi i kēia mau waiwai i loko o ke ʻano kihi:
\[ \cos \theta = \frac{\mathbf{H} \cdot \mathbf{I}}{\| \mathbf{H} \| \| \mathbf{I} \|} = \frac{8}{7\sqrt{21}} \]
\[ \theta = \cos^{-1} \left( \frac{8}{7\sqrt{21}} \right) \]
ʻO ka hopena hope loa, hiki iā mākou ke hoʻohana i ka mīkini helu e ʻike i ka waiwai o ke kihi:
\[ \theta \approx 73,4^\circ \]
Ka hopena
He mea koʻikoʻi ke kumumanaʻo o nā hana vector i ka makemakika a me ka ʻepekema. Kūkākūkā kēia ʻatikala i kekahi mau pilikia hoʻohālike a me kā lākou mau hoʻonā, mai ka hoʻohui a me ka hoʻemi ʻana o ka vector, ka hoʻonui scalar, ka huahana kiko, ka huahana kea, a me ka hoʻoholo ʻana i ke kihi ma waena o ʻelua mau vector. Ma ka hana ʻana ma o kēia mau hoʻohālike, manaʻolana mākou e hoʻonui i kou ʻike i nā hana vector a kōkua iā ʻoe e hoʻoponopono i nā pilikia e pili ana i nā vectors ma nā ʻano like ʻole.
