Fa'ata'ita'iga o se Fesili Talanoaga i le Vector Iunite o se Vector
Pendahuluan
I le matematika ma le fisiki, o vectors o elemene taua ia e fai ma sui o le tele ma le itu. E masani ona faʻaaogaina vectors e faʻamatalaina ai mea eseese e pei o le saoasaoa, malosi, ma le fesiitaiga i le avanoa e lua pe tolu-dimensional. O se tasi o manatu taua e fesoʻotaʻi ma vectors o le unit vector. O lenei tusiga o le a talanoaina ai le faʻamatalaga o se unit vector, le auala e fuafua ai, ma tuʻuina atu ni faʻataʻitaʻiga o faʻafitauli ma fofo.
Malamalama i Vekita o Iunite
O le vector iunite o se vector e tasi le iunite lona tele ma e tutusa le itu ma le vector muamua. E masani ona faʻaaogaina vector iunite e faʻafaigofie ai le auʻiliʻiliga aua e tasi lava lo latou tele, ma mafai ai ona taulaʻi atu le taulaʻiga autu i lo latou itu. Ina ia liua se vector i se vector iunite, e tatau ona tatou vaevaeina ona vaega taʻitasi i le tele o le vector.
I le fa'amatematika, afai o le \( \mathbf{v} \) o se vector, ona mafai lea ona fa'aalia lona vector iunite \( \mathbf{\hat{v}} \) e pei ona fa'amatalaina:
\[
\mathbf{\hat{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}
\]
lea o le \( \|\mathbf{v}\| \) o le tele po'o le umi o le vector \( \mathbf{v} \).
Fuafuaina o le Tele o le Vector
E mafai ona fuafuaina le telē o se vector \( \mathbf{v} \) i le avanoa lua-dimensional faatasi ai ma vaega \( (v_x, v_y) \) e faaaoga ai le fua faatatau:
\[
\|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2}
\]
I le taimi nei, mo vectors i le avanoa tolu-dimensional ma vaega \((v_x, v_y, v_z) \), o le tele e fuafuaina e faʻaaoga ai le fua faʻatatau:
\[
\|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2 + v_z^2}
\]
Fesili Fa'ata'ita'i ma Talanoaga
Mo se faamaninoga o le manatu o unit vectors, se'i o tatou tilotilo i ni fa'ata'ita'iga o fesili ma a latou talanoaga.
Fa'ata'ita'iga Fesili 1
Fesili: Tuuina atu se vector \( \mathbf{a} = (3, 4) \). Fuafua le vector iunite o le vector \( \mathbf{a} \).
Talanoaga:
1. Fuafua vaega o le vector \( \mathbf{a} \):
\( a_x = 3 \), \( a_y = 4 \)
2. Fuafua le telē o le vector \( \mathbf{a} \):
\[
\|\mathbf{a}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
3. Fuafua le vector iunite e ala i le vaevaeina o vaega taʻitasi o le vector \( \mathbf{a} \) i lona tele:
\[
\mathbf{\hat{a}} = \left( \frac{3}{5}, \frac{4}{5} \right) = \left( 0.6, 0.8 \right)
\]
O lea la, o le vector iunite o le \( \mathbf{a} \) o le \( (0.6, 0.8) \).
Fa'ata'ita'iga Fesili 2
Fesili: Tuuina atu se vector \( \mathbf{b} = (1, -2, 2) \). Fuafua le vector iunite o le vector \( \mathbf{b} \).
Talanoaga:
1. Fuafua vaega o le vector \( \mathbf{b} \):
\( b_x = 1 \), \( b_y = -2 \), \( b_z = 2 \)
2. Fuafua le telē o le vector \( \mathbf{b} \):
\[
\|\mathbf{b}\| = \sqrt{1^2 + (-2)^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3
\]
3. Fuafua le vector iunite e ala i le vaevaeina o vaega taʻitasi o le vector \( \mathbf{b} \) i lona tele:
\[
\mathbf{\hat{b}} = \left( \frac{1}{3}, \frac{-2}{3}, \frac{2}{3} \right) \approx \left( 0.333, -0.667, 0.667 \right)
\]
O lea la, o le vector iunite o le \( \mathbf{b} \) o le \( \left( 0.333, -0.667, 0.667 \right) \).
Fa'ata'ita'iga Fesili 3
Fesili: Tuuina atu le vector \( \mathbf{c} = (-7, 24) \). Fuafua le vector iunite o le vector \( \mathbf{c} \).
Talanoaga:
1. Fuafua vaega o le vector \( \mathbf{c} \):
\( c_x = -7 \), \( c_y = 24 \)
2. Fuafua le telē o le vector \( \mathbf{c} \):
\[
\|\mathbf{c}\| = \sqrt{(-7)^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25
\]
3. Fuafua le vector iunite e ala i le vaevaeina o vaega taʻitasi o le vector \( \mathbf{c} \) i lona tele:
\[
\mathbf{\hat{c}} = \left( \frac{-7}{25}, \frac{24}{25} \right) = \left( -0.28, 0.96 \right)
\]
O lea la, o le vector iunite o le \( \mathbf{c} \) o le \( (-0.28, 0.96) \).
Fa'ata'ita'iga Fesili 4
Fesili: Afai o se vector \( \mathbf{d} = (6, 8, 0) \), fuafua le vector iunite o le vector \( \mathbf{d} \).
Talanoaga:
1. Fuafua vaega o le vector \( \mathbf{d} \):
\( d_x = 6 \), \( d_y = 8 \), \( d_z = 0 \)
2. Fuafua le telē o le vector \( \mathbf{d} \):
\[
\|\mathbf{d}\| = \sqrt{6^2 + 8^2 + 0^2} = \sqrt{36 + 64 + 0} = \sqrt{100} = 10
\]
3. Fuafua le vector iunite e ala i le vaevaeina o vaega taʻitasi o le vector \( \mathbf{d} \) i lona tele:
\[
\mathbf{\hat{d}} = \left( \frac{6}{10}, \frac{8}{10}, \frac{0}{10} \right) = \left( 0.6, 0.8, 0 \right)
\]
O lea la, o le vector iunite o le \( \mathbf{d} \) o le \( (0.6, 0.8, 0) \).
Fa'ato'a
E ala i le talanoaga ma faʻataʻitaʻiga o loʻo i luga, e mafai ona tatou malamalama o le fuafuaina o se vector iunite e manaʻomia ai le fuafuaina o le tele o le vector ona vaevaeina lea o vaega o le vector i lena tele. E matua aoga tele vector iunite i le tele o faʻaoga e pei o le faʻatulagaina o vector i ata komepiuta, auiliiliga o le malosi i le fisiki, ma le tele o isi matāʻupu. I le malamalama i lenei manatu, e tatau ona faigofie ona tatou taulimaina faʻafitauli e aofia ai vectors.
