Fa'ata'ita'iga o Fesili Talanoaga o le Fa'agaioiga o Vector
O galuega fa'atino vector o se manatu fa'avae i le matematika e masani ona alia'e mai i vaega eseese o su'esu'ega, e pei o le fisiki, inisinia, ma le saienisi komepiuta. I totonu o lenei tusiga, o le a tatou talanoaina ai ni fa'ata'ita'iga o galuega fa'atino vector ma a latou fofo e maua ai se malamalamaga loloto ma sili atu ona mautu. O nei fa'ata'ita'iga o le a aofia ai galuega fa'atino fa'avae e pei o le fa'aopoopoga ma le to'esea o vector, fa'apea fo'i ma galuega fa'atino e sili atu ona alualu i luma e pei o le fa'ateleina scalar ma le fa'ateleina cross-vector.
1. Fa'aopoopoga ma To'esega o le Vector
Fa'ata'ita'iga Fesili 1
Tuuina atu ni vectors se lua A ma le B i le tulaga o le component:
\[ \mathbf{A} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} \]
\[ \mathbf{B} = \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} \]
Fuafua le iʻuga o le faʻaopoopoga ma le toʻesega o vectors e lua.
Talanoaga
Mo le fa'aopoopoga vector, tatou te fa'aopoopoina vaega ta'itasi e fetaui ma vectors e lua.
\[ \mathbf{A} + \mathbf{B} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} + \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 + (-1) \\ 3 + 4 \\ -1 + 2 \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \\ 1 \end{pmatrix} \]
Mo le to'esega o vector, matou te to'esea vaega ta'itasi o vectors uma e lua.
\[ \mathbf{A} – \mathbf{B} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} – \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 – (-1) \\ 3 – 4 \\ -1 – 2 \end{pmatrix} = \begin{pmatrix} 3 \\ -1 \\ -3 \end{pmatrix} \]
2. Fa'atelega Fa'asolosolo i le Vector
Fa'ata'ita'iga Fesili 2
I le tuuina atu o le vector C ma le scalar k:
\[ \mathbf{C} = \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix} \]
\[ k = 4 \]
Fuafua le fua fa'atatau o le vector C i le scalar k.
Talanoaga
O le fa'ateleina o se scalar i se vector e faia e ala i le fa'ateleina o vaega ta'itasi o le vector i le 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. Oloa Fa'ailoga
Fa'ata'ita'iga Fesili 3
Tuuina atu ni vectors se lua D ma le E:
\[ \mathbf{D} = \begin{pmatrix} 3 \\ -2 \\ 4 \end{pmatrix} \]
\[ \mathbf{E} = \amata{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix} \]
Fuafua le dot product o vectors e lua.
Talanoaga
E maua le dot product (OF) o vectors e lua e ala i le faaopoopoina o oloa o a la vaega tutusa.
\[ \mathbf{D} \cdot \mathbf{E} = 3 \cdot 1 + (-2) \cdot 0 + 4 \cdot (-1) = 3 + 0 – 4 = -1 \]
4. Oloa Fa'alava
Fa'ata'ita'iga Fesili 4
Tuuina atu ni vectors se lua F ma le G:
\[ \mathbf{F} = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} \]
\[ \mathbf{G} = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix} \]
Fuafua le fa'asologa o le oloa (cross product) o vectors e lua.
Talanoaga
O le oloa fa'alava o vectors e lua i le avanoa tolu-dimensional e maua e ala i le fa'aogaina o le determinant o le matrix na fausia e na vectors. O le oloa fa'alava e tu'uina atu e le fua fa'atatau:
\[ \mathbf{F} \times \mathbf{G} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 3 & 4 \\ 1 & -1 & 2 \end{vmatrix} \]
E mafai ona fuafuaina lenei mea i le auala lenei:
\[
\mathbf{F} \times \mathbf{G} = \mathbf{i} \begin{vmatrix} 3 & 4 \\ -1 & 2 \end{vmatrix} – \mathbf{j} \begin{vmatrix} 2 & 4 \\ 1 & 2 \end{vmatrix} {k} 3 & -1 \end{vmatrix}
\]
Fuafuaina o le mea e fuafua ai vaega taʻitasi o le submatrix:
\[
= \mathbf{i} (3 \cdot 2 – 4 \cdot -1) – \mathbf{j} (2 \cdot 2 – 4 \cdot 1) + \mathbf{k} (2 \cdot -1 – 3 \cdot 1)
\]
\[
= \mathbf{i} (6 + 4) – \mathbf{j} (4 – 4) + \mathbf{k} (-2 – 3)
\]
\[
= \mathbf{i} (10) – \mathbf{j} (0) + \mathbf{k} (-5)
\]
\[
= \begin{pmatrix} 10 \\ 0 \\ -5 \end{pmatrix}
\]
O lea la, o le oloa fa'alava o le F ma le G e:
\[ \mathbf{F} \times \mathbf{G} = \begin{pmatrix} 10 \\ 0 \\ -5 \end{pmatrix} \]
5. Fuafuaina o le tulimanu i le va o Vectors e lua
Fa'ata'ita'iga Fesili 5
Tuuina atu ni vectors se lua H ma le I:
\[ \mathbf{H} = \begin{pmatrix} 6 \\ 2 \\ 3 \end{pmatrix} \]
\[ \mathbf{I} = \begin{pmatrix} 1 \\ 4 \\ -2 \end{pmatrix} \]
Fuafua le tulimanu i le va o vectors e lua.
Talanoaga
E mafai ona maua le tulimanu \(\theta\) i le va o vectors e lua e ala i le fa'aogaina o le sootaga i le va o le dot product ma le tele o vectors e lua:
\[ \mathbf{H} \cdot \mathbf{I} = \| \mathbf{H} \| \| \mathbf{I} \| \cos \theta \]
Muamua, fuafua le fua fa'atatau o le dot product \( \mathbf{H} \cdot \mathbf{I} \):
\[ \mathbf{H} \cdot \mathbf{I} = 6 \cdot 1 + 2 \cdot 4 + 3 \cdot (-2) = 6 + 8 – 6 = 8 \]
Sosoo ai, fuafua le telē o vectors uma e lua:
\[ \| \mathbf{H} \| = \sqrt{6^2 + 2^2 + 3^2} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]
\[ \| \mathbf{I} \| = \sqrt{1^2 + 4^2 + (-2)^2} = \sqrt{1 + 16 + 4} = \sqrt{21} \]
Ona sui lea o nei tau i le fua fa'atatau o le tulimanu:
\[ \cos \theta = \frac{\mathbf{H} \cdot \mathbf{I}}{\| \mathbf{H} \| \| \mathbf{I} \|} = \frac{8}{7\sqrt{21}} \]
\[ \theta = \cos^{-1} \left( \frac{8}{7\sqrt{21}} \right) \]
O le iʻuga mulimuli, e mafai ona tatou faʻaaogaina le calculator e suʻe ai le tau o le tulimanu:
\[ \theta \pe tusa ma le 73,4^\circ \]
I'uga
O le manatu o galuega fa'atino vector e taua tele i le matematika ma le saienisi. O lenei tusiga o lo'o talanoaina ai ni fa'ata'ita'iga o fa'afitauli ma a latou fofo, e amata mai i le fa'aopoopoina ma le to'esea o vector, fa'atelega scalar, dot product, cross product, ma le fuafuaina o le tulimanu i le va o vectors e lua. I le galulue fa'atasi ma nei fa'ata'ita'iga, matou te fa'amoemoe e fa'aleleia atili lou malamalama i galuega fa'atino vector ma fesoasoani ia te oe e foia fa'afitauli e aofia ai vectors i tulaga eseese.
