Mehlala ea Lipotso Tse Buisanang ka Li-Vector Tse Lekanang Tsamaisong ea Cartesian Coordinate
Pendahuluan
Lipalong, vektara ke ntho e nang le boholo le tataiso. Livektara li na le lits'ebetso mafapheng a fapaneng a kang fisiks, boenjiniere le saense ea khomphutha. Sehloohong sena, re tla tšohla mohopolo oa livektara tse lekanang tsamaisong ea li-coordinate tsa Cartesian le ho hlahisa mehlala le litharollo. Ho utloisisa livektara tse lekanang ho bohlokoa lits'ebetsong tse fapaneng, ho kenyeletsoa le mechini le litšoantšo tsa khomphutha.
Metheo ea Li-vector Sistemeng ea Cartesian Coordinate
Sistimi ea ho hokahanya ea Cartesian ke sistimi ea mahlakore a mabeli e nang le li-axes tsa X le Y tse otlolohileng. Sistimi ena, li-vector hangata li emeloa e le lipara tse laetsoeng (x, y), moo x le y e leng likarolo tsa vector ho latela li-axes tsa X le Y, ka ho latellana.
A re re re na le lintlha tse peli tsamaisong ea likhokahano tsa Cartesian, \(A(x_1, y_1)\) le \(B(x_2, y_2)\). Vekthara e hokahanyang lintlha tsena tse peli e ka hlalosoa e le \( \vec{AB} = (x_2 – x_1, y_2 – y_1) \).
Li-vector tse lekanang
Ho thoe li-vector tse peli lia lekana haeba li na le boholo le tataiso e tšoanang. Ho ea ka lipalo, li-vector tse peli \( \vec{u} = (u_1, u_2) \) le \( \vec{v} = (v_1, v_2) \) lia lekana haeba le haeba feela:
\[
\vec{u} = \vec{v} \quad \text{or} \quad (u_1 = v_1 \text{ le } u_2 = v_2)
\]
Sena se bolela hore likarolo tse tsamaellanang tsa li-vector tse peli li tlameha ho tšoana.
Lipotso tsa Mehlala le Puisano
Potso ea 1: Ho Fumana Livekthara tse Lekanang
Ho fanoe ka lintlha tse tharo tsamaisong ea likhokahano tsa Cartesian: \( A(2, 3) \), \( B(5, 7) \), le \( C(7, -1) \). Fumana hore na vekthara \( \vec{AB} \) e lekana le vekthara \( \vec{AC} \).
Puisano:
– Fumana vekthara \( \vec{AB} \):
\[
\vec{AB} = (5 – 2, 7 – 3) = (3, 4)
\]
– Fumana vekthara \( \vec{AC} \):
\[
\vec{AC} = (7 – 2, -1 – 3) = (5, -4)
\]
Kamora ho bala dikarolo tsa vekthara ka nngwe, re bona hore \( \vec{AB} = (3, 4) \) le \( \vec{AC} = (5, -4) \). Kaha \( (3, 4) \neq (5, -4) \), vekthara \( \vec{AB} \) ha e lekane le vekthara \( \vec{AC} \).
Potso ea 2: Ho aha livektha tse lekanang
Fumana ntlha \( D \) e le hore vekthara \( \vec{AB} = \vec{CD} \) e nang le ntlha \( C(4, -2) \), ntlha \( B(8, 3) \), le \( A(2, 1) \).
Puisano:
– Fumana vekthara \( \vec{AB} \):
\[
\vec{AB} = (8 – 2, 3 – 1) = (6, 2)
\]
Kaha \( \vec{CD} \) e tlameha ho lekana le \( \vec{AB} \), joale:
\[
\vec{CD} = \vec{AB} = (6, 2)
\]
– A re re \( D(x, y) \). Ebe \( \vec{CD} = (x – 4, y + 2) \). Ho tloha mona re fumana:
\[
(x – 4, y + 2) = (6, 2)
\]
Ka ho lekanya likarolo tse loketseng, re fumana:
\[
x – 4 = 6 \quad \Rightmouse \quad x = 10
\]
\[
y + 2 = 2 \quad \Rightarrow \quad y = 0
\]
Kahoo, ntlha \( D \) ke \( (10, 0) \).
Potso ea 3: Bopaki bo nang le boholo ba vektheri
Paka hore divekthara \( \vec{PQ} \) le \( \vec{RS} \) di a lekana, ha ho fanwa \( P(1, 2) \), \( Q(4, 6) \), \( R(-3, -7) \), le \( S(0, -3) \).
Puisano:
– Fumana vekthara \( \vec{PQ} \):
\[
\vec{PQ} = (4 – 1, 6 – 2) = (3, 4)
\]
– Hlalosa vekthara \( \vec{RS} \):
\[
\vec{RS} = (0 – (-3), -3 – (-7)) = (3, 4)
\]
Ho tsoa liphethong tsa lipalo, re bona hore \( \vec{PQ} = (3, 4) \) le \( \vec{RS} = (3, 4) \). Kaha livekthara ka bobeli li na le likarolo tse tšoanang, \( \vec{PQ} \) e lekana le \( \vec{RS} \).
Tšebeliso ea Li-Vector tse Lekanang
Li-vector tse lekanang li sebelisoa khafetsa mafapheng a fapaneng a saense. Fisiks, li sebelisoa ho hlalosa matla kapa ho falla ho nang le boholo le tataiso e tšoanang. Litšoantšong tsa khomphutha, li-vector li sebelisoa ho fetola le ho phelisetsa lintho tsa litšoantšo ka katleho.
Qetello
Ho utloisisa mohopolo oa li-vector tse lekanang tsamaisong ea li-coordinate tsa Cartesian ke motheo oa bohlokoa bakeng sa lipalo le ts'ebeliso ea tsona e pharaletseng. Sengoloa sena se buile ka mokhoa oa ho fumana li-vector tse lekanang ka mathata a 'maloa a mehlala le litharollo tsa ona. Ka ho utloisisa le ho sebelisa mohopolo ona, re ka rarolla mathata a fapaneng a kenyeletsang tlhahlobo ea li-vector masimong a mangata a mahlale.
Re tšepa hore puisano ena e tla u thusa ho utloisisa mohopolo oa li-vector tse lekanang tsamaisong ea li-coordinate tsa Cartesian. Thuto e monate, le mahlohonolo ho tsebeng li-vector hantle!
