Mohlala oa Potso ea Puisano mabapi le Vektara ea Yuniti ea Vektara
Pendahuluan
Lipalong le fisiks, li-vector ke likarolo tsa motheo tse emelang boholo le tataiso. Li-vector hangata li sebelisoa ho hlalosa liketsahalo tse fapaneng tse kang lebelo, matla le ho falla sebakeng sa mahlakore a mabeli kapa a mararo. Khopolo e 'ngoe ea bohlokoa e amanang le li-vector ke vector ea yuniti. Sehlooho sena se tla tšohla tlhaloso ea vector ea yuniti, mokhoa oa ho e bala, le ho fana ka mehlala e 'maloa ea mathata le litharollo.
Ho Utloisisa Li-Vector tsa Yuniti
Vekthara ea yuniti ke vekthara e nang le boholo ba yuniti e le 'ngoe le tataiso e tšoanang le vekthara ea pele. Livekthara tsa yuniti hangata li sebelisoa ho nolofatsa tlhahlobo hobane boholo ba tsona kamehla ke ntho e le 'ngoe, e leng se lumellang hore sepheo sa mantlha se be tataisong ea tsona. Ho fetolela vekthara ho ba vekthara ea yuniti, re tlameha ho arola karolo ka 'ngoe ea eona ka boholo ba vekthara.
Ho ya ka dipalo, haeba \( \mathbf{v} \) e le vekthara, vekthara ya yona ya yuniti \( \mathbf{\hat{v}} \) e ka hlaloswa ka tsela ena:
\[
\mathbf{\hat{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}
\]
moo \( \|\mathbf{v}\| \) e leng boholo kapa bolelele ba vekthara \( \mathbf{v} \).
Ho Bala Boholo ba Vekthara
Boholo ba vekthara \( \mathbf{v} \) sebakeng sa mahlakore a mabedi se nang le dikarolo \( (v_x, v_y) \) bo ka balwa ho sebediswa foromo:
\[
\|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2}
\]
Ho sa le jwalo, bakeng sa divekthara tse sebakeng sa mahlakore a mararo tse nang le dikarolo \( (v_x, v_y, v_z) \), boholo bo balwa ho sebediswa foromo:
\[
\|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2 + v_z^2}
\]
Lipotso tsa Mehlala le Puisano
Ho hlakisa mohopolo oa li-vector tsa yuniti, ha re shebeng mehlala e meng ea lipotso le lipuisano tsa tsona.
Mohlala oa Potso ea 1
Potso: Fanoe ka vekthara \( \mathbf{a} = (3, 4) \). Fumana vekthara ea yuniti ea vekthara \( \mathbf{a} \).
Puisano:
1. Fumana likarolo tsa vekthara \( \mathbf{a} \):
\( a_x = 3 \), \( a_y = 4 \)
2. Bala boholo ba vekthara \( \mathbf{a} \):
\[
\|\mathbf{a}\| = \ sqrt{3^2 + 4^2} = \ sqrt{9 + 16} = \ sqrt{25} = 5
\]
3. Bala vekthara ea yuniti ka ho arola karolo e 'ngoe le e 'ngoe ea vekthara \( \mathbf{a} \) ka boholo ba eona:
\[
\mathbf{\hat{a}} = \left( \frac{3}{5}, \frac{4}{5} \right) = \left( 0.6, 0.8 \right)
\]
Kahoo, vekthara ya yuniti ya \( \mathbf{a} \) ke \( (0.6, 0.8) \).
Mohlala oa Potso ea 2
Potso: Fanoeng ka vekthara \( \mathbf{b} = (1, -2, 2) \). Fumana vekthara ea yuniti ea vekthara \( \mathbf{b} \).
Puisano:
1. Fumana likarolo tsa vekthara \( \mathbf{b} \):
\( b_x = 1 \), \( b_y = -2 \), \( b_z = 2 \)
2. Bala boholo ba vekthara \( \mathbf{b} \):
\[
\|\mathbf{b}\| = \sqrt{1^2 + (-2)^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3
\]
3. Bala vekthara ea yuniti ka ho arola karolo e 'ngoe le e 'ngoe ea vekthara \( \mathbf{b} \) ka boholo ba eona:
\[
\mathbf{\hat{b}} = \left( \frac{1}{3}, \frac{-2}{3}, \frac{2}{3} \right) \approx \left( 0.333, -0.667, 0.667 \right)
\]
Kahoo, vekthara ya yuniti ya \( \mathbf{b} \) ke \( \left( 0.333, -0.667, 0.667 \right) \).
Mohlala oa Potso ea 3
Potso: Ha ho fanoe ka vekthara \( \mathbf{c} = (-7, 24) \). Fumana vekthara ea yuniti ea vekthara \( \mathbf{c} \).
Puisano:
1. Fumana likarolo tsa vekthara \( \mathbf{c} \):
\( c_x = -7 \), \( c_y = 24 \)
2. Bala boholo ba vekthara \( \mathbf{c} \):
\[
\|\mathbf{c}\| = \ sqrt{(-7)^2 + 24^2} = \ sqrt{49 + 576} = \ sqrt{625} = 25
\]
3. Bala vekthara ea yuniti ka ho arola karolo e 'ngoe le e 'ngoe ea vekthara \( \mathbf{c} \) ka boholo ba eona:
\[
\mathbf{\hat{c}} = \left( \frac{-7}{25}, \frac{24}{25} \right) = \left( -0.28, 0.96 \right)
\]
Kahoo, vekthara ya yuniti ya \( \mathbf{c} \) ke \( (-0.28, 0.96) \).
Mohlala oa Potso ea 4
Potso: Haeba vekthara \( \mathbf{d} = (6, 8, 0) \), fumana vekthara ya yuniti ya vekthara \( \mathbf{d} \).
Puisano:
1. Fumana likarolo tsa vekthara \( \mathbf{d} \):
\( d_x = 6 \), \( d_y = 8 \), \( d_z = 0 \)
2. Bala boholo ba vekthara \( \mathbf{d} \):
\[
\|\mathbf{d}\| = \sqrt{6^2 + 8^2 + 0^2} = \sqrt{36 + 64 + 0} = \sqrt{100} = 10
\]
3. Bala vekthara ea yuniti ka ho arola karolo e 'ngoe le e 'ngoe ea vekthara \( \mathbf{d} \) ka boholo ba eona:
\[
\mathbf{\hat{d}} = \left( \frac{6}{10}, \frac{8}{10}, \frac{0}{10} \right) = \left( 0.6, 0.8, 0 \right)
\]
Kahoo, vekthara ya yuniti ya \( \mathbf{d} \) ke \( (0.6, 0.8, 0) \).
Ho koala
Ka puisano le mehlala e kaholimo, re ka utloisisa hore ho bala vektara ea yuniti ho hloka ho bala boholo ba vektara ebe ho arola likarolo tsa vektara ka boholo boo. Livektara tsa yuniti li na le thuso haholo lits'ebetsong tse fapaneng tse kang ho tloaeleha ha vektara litšoantšong tsa khomphutha, tlhahlobo ea matla fisiks, le masimo a mang a mangata. Ka ho utloisisa mohopolo ona, re lokela ho khona ho sebetsana le mathata a amanang le livektara habonolo.
