Fa'ata'ita'iga o Fesili e Talanoaina ai Vectors ma a Latou Galuega
O vectors o se manatu faavae i le matematika ma le fisiki, e masani ona faʻaaogaina i vaega faasaienisi eseese. O vectors e fai ma sui o aofaʻiga i le tele ma le itu. O loʻo i lalo nisi o faʻataʻitaʻiga o faʻafitauli e aofia ai vectors ma talanoaga o galuega eseese e pei o le faʻaopoopoga, toʻesega, ma le faʻateleina i scalars. O lenei tusiga o le a tuʻuina atu ai se malamalamaga loloto i le auala e foia ai faʻafitauli e aofia ai vectors.
1. Fa'aopoopoga Vekita
Fa'ata'ita'iga Fesili 1
Tuuina atu ni vectors se lua i le tulaga o vaega:
A = (3, 4)
B = (1, 2)
Fuafua le iʻuga o le faaopoopoina o vectors A ma le B.
Talanoaga
E faia le fa'aopoopoga o vector e ala i le fa'aopoopoina o vaega talafeagai o vectors e lua. O lea la, e mafai ona tatou fuafuaina
\[
A + B = (3 + 1, 4 + 2) = (4, 6)
\]
O lea la, o le taunuuga o le faaopoopoina o vectors A ma le B o le (4, 6).
2. To'esega o le Veka
Fa'ata'ita'iga Fesili 2
Tuuina atu ni vectors se lua i le tulaga o vaega:
C = (5, 7)
D = (2, 3)
Fuafua le iʻuga o le toʻesea o le vector C mai le vector D.
Talanoaga
E faia le to'esea o vector e ala i le to'esea o vaega talafeagai o vectors e lua. O lea la, e mafai ona tatou fuafuaina
\[
C – D = (5 – 2, 7 – 3) = (3, 4)
\]
O lea la, o le taunuuga o le to'esea o vectors C ma le D o le (3, 4).
3. Fa'atelega o Vectors i Scalars
Fa'ata'ita'iga Fesili 3
Tuuina atu le vector E = (4, -2) ma le scalar k = 3. Fuafua le iʻuga o le faʻateleina o le vector E i le scalar k.
Talanoaga
O le fa'ateleina o se vector i se scalar e faia e ala i le fa'ateleina o vaega ta'itasi o le vector i le scalar. O lea la, e mafai ona tatou fuafuaina
\[
k E = 3 (4, -2) = (3 4, 3 -2) = (12, -6)
\]
O lea la, o le taunuuga o le faateleina o le vector E i le scalar k o le (12, -6).
4. Oloa Fa'ailoga
Fa'ata'ita'iga Fesili 4
Tuuina atu ni vectors se lua i le tulaga o vaega:
F = (1, 3)
G = (4, 2)
Fuafua le dot product o vectors F ma le G.
Talanoaga
O le dot product o vectors e lua o le aofaʻi lea o oloa o a la vaega tutusa. O lea la, e mafai ona tatou fuafuaina
\[
F \cdot G = (1 4) + (3 2) = 4 + 6 = 10
\]
O lea la, o le dot product o vectors F ma le G e 10.
5. Oloa Fa'alava
Fa'ata'ita'iga Fesili 5
Tuuina atu ni vectors se lua i le 3D:
H = (2, -3, 1)
A'u = (1, 4, -2)
Fuafua le fa'asologa o le oloa (cross product) o vectors H ma le I.
Talanoaga
O le oloa fa'alava o vectors e lua e 3-dimensional e gaosia e le determinant o le matrix o lo'o i ai vaega o vectors uma e lua. O le vector returned e iai vaega nei:
\[
H \times I = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
2 & -3 & 1 \\
1 & 4 & -2 \\
\end{vmatrix}
\]
I le fuafuaina o le determinant, matou te maua:
\[
H taimi I = (\mathbf{i}((-3)(-2) – (1)(4)) – \mathbf{j}(2(-2) – (1)(1)) + \mathbf{k}(2(4) – (-3)(1)))
\]
\[
= (\mathbf{i}(6 – 4) – \mathbf{j}(-4 – 1) + \mathbf{k}(8 + 3))
\]
\[
= (\mathbf{i}(2) – \mathbf{j}(-5) + \mathbf{k}(11))
\]
\[
= (2, 5, 11)
\]
O lea la, o le fua fa'atatau o vectors H ma le I o le (2, 5, 11).
6. Umi po'o le Tele o le Vekita
Fa'ata'ita'iga Fesili 6
Afai ua tu'uina atu se vector J = (6, 8). Fuafua le umi (tele) o le vector J.
Talanoaga
O le umi (tele) o se vector e fuafuaina e faʻaaoga ai le fua faʻatatau:
\[
\| J \| = \sqrt{x^2 + y^2}
\]
I lenei tulaga, \( x = 6 \) ma le \( y = 8 \), ina ia:
\[
\| J \| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]
O lea la, o le umi (tele) o le vector J e 10.
7. Veketo o le Iunite
Fa'ata'ita'iga Fesili 7
Afai ua tu'uina atu se vector K = (-5, 12). Saili le vector iunite o le K.
Talanoaga
O le vector iunite o se vector e 1 lona umi. Ina ia maua le vector iunite o se vector, e tatau ona tatou vaevaeina vaega taʻitasi o le vector i le umi (tele) o le vector. E mafai ona fuafuaina le umi o le vector K e pei ona taua i lalo:
\[
\| K \|= \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]
Ona o le vector iunite K lea:
\[
\hat{K} = \left(\frac{-5}{13}, \frac{12}{13}\right)
\]
O lea la, o le vector iunite o le vector K o le \(\left(\frac{-5}{13}, \frac{12}{13}\right)\).
I'uga
E ala i faʻataʻitaʻiga o loʻo i luga, ua tatou vaʻaia ai le auala e galue ai vectors ma a latou galuega i tulaga eseese. O le faʻaopoopoga ma le toʻesea o vectors e aofia ai le faʻaopoopoina ma le toʻesea o vaega talafeagai. E mafai ona faia le faʻateleina o vectors i le scalar poʻo le dot product form, ma i le cross product form mo vectors 3D. E mafai foʻi ona tatou fuafuaina le umi o se vector ma maua lona unit vector.
E taua tele le malamalama i nei manatu faavae auā e faʻaaogaina vectors i le tele o faʻaoga i le tele o vaega, e aofia ai le fisiki, inisinia, ma ata komepiuta. Afai e lava le faʻataʻitaʻi, e mafai ona tatou faʻataʻitaʻiina nei galuega ma faʻaaoga i auʻiliʻiliga ma le foʻia o faʻafitauli e sili atu ona faigata.
