Mehlala ea Lipotso Tse Buisanang ka Li-Vector Tse Fosahetseng kapa Li-Vector Tse Fapaneng
Lipalong, haholo-holo fisiks kapa jeometri ea tlhahlobo, khopolo ea li-vector e bapala karolo ea bohlokoa. Li-vector hangata li sebelisoa ho emela bongata ka tataiso le boholo, joalo ka lebelo, matla le ho falla. Ha re bua ka li-vector, hangata re kopana le mantsoe "vector e mpe" kapa "vector e fapaneng." Sengoloa sena se tla hlalosa khopolo ena ka botebo le ho fana ka mehlala le litharollo ho nolofatsa kutloisiso.
Tlhaloso ea Vektha e Negative
Vekthara e mpe, kapa vekthara e fapaneng, ke vekthara e nang le lehlakore le fapaneng empa boholo bo lekanang le vekthara ea pele. Haeba re na le vekthara \(\mathbf{a}\), joale vekthara e mpe ea \(\mathbf{a}\), hangata e hlalosoang e le \(-\mathbf{a}\), e na le lehlakore le fapaneng le boholo bo tšoanang le \(\mathbf{a}\). Haeba \(\mathbf{a}\) e emetsoe ka sebopeho sa karolo e le \((a_x, a_y)\), joale vekthara e mpe ke \((-a_x, -a_y)\).
Mongolo oa Vector le Boemeli
A re re vekthara \(\mathbf{a}\) e emetswe ka sebopeho sa karolo e le:
\[ \mathbf{a} = a_x \mathbf{i} + a_y \mathbf{j} \]
moo \(\mathbf{i}\) le \(\mathbf{j}\) e leng divekthara tsa yuniti ka tsela ya x- le y, ka ho latellana. Ebe, vekthara e mpe \(\mathbf{a}\) kapa \(-\mathbf{a}\) e ka emelwa jwalo ka:
\[ -\mathbf{a} = -a_x \mathbf{i} – a_y \mathbf{j} \]
Matlotlo a Li-vector tse mpe
Litšobotsi tse ling tsa bohlokoa tsa li-vector tse mpe li kenyelletsa:
1. Ho eketsa ka Vekthara ea Pele: Ho eketsa vekthara ka vekthara ea eona e mpe ho tla hlahisa vekthara ea lefela.
\[ \mathbf{a} + (-\mathbf{a}) = \mathbf{0} \]
2. Ts'ebetso ea Scalar: Ho atisa vekthara ka -1 ho tla hlahisa vekthara ea eona e mpe.
\[ -1 \cdot \mathbf{a} = -\mathbf{a} \]
Lipotso tsa Mehlala le Puisano
Ho utloisisa hamolemo mohopolo oa li-vector tse mpe kapa li-vector tse hanyetsanang, ha re sebetseng mathateng a latelang a mehlala:
Mohlala oa 1:
A re re ho na le vekthara \(\mathbf{a} = 3 \mathbf{i} – 4 \mathbf{j}\). Fumana vekthara e mpe ea vekthara \(\mathbf{a}\).
Puisano:
Hoa tsebahala:
\[ \mathbf{a} = 3 \mathbf{i} – 4 \mathbf{j} \]
Vekthara e mpe ea \(\mathbf{a}\) ke:
\[ -\mathbf{a} = -1 \cdot (3 \mathbf{i} – 4 \mathbf{j}) \]
\[ -\mathbf{a} = -3 \mathbf{i} + 4 \mathbf{j} \]
Kahoo, vekthara e mpe ea \(\mathbf{a}\) ke:
\[ -\mathbf{a} = -3 \mathbf{i} + 4 \mathbf{j} \]
Mohlala oa 2:
Ho na le li-vector tse peli \(\mathbf{b} = 6 \mathbf{i} + 2 \mathbf{j}\) le \(\mathbf{c} = -1 \mathbf{i} + 7 \mathbf{j}\). Fumana sehlahisoa sa \(\mathbf{b} + (-\mathbf{c})\).
Puisano:
Hoa tsebahala:
\[ \mathbf{b} = 6 \ lipalobf{i} + 2 \ lipalo{j} \]
\[ \mathbf{c} = -1 \mathbf{i} + 7 \mathbf{j} \]
Vekthara e mpe ea \(\mathbf{c}\) ke:
\[ -\mathbf{c} = -1 \cdot (-1 \mathbf{i} + 7 \mathbf{j}) \]
\[ -\mathbf{c} = 1 \mathbf{i} – 7 \mathbf{j} \]
Jwale re fumana \(\mathbf{b} + (-\mathbf{c})\):
\[ \mathbf{b} + (-\mathbf{c}) = (6 \mathbf{i} + 2 \mathbf{j}) + (1 \mathbf{i} – 7 \mathbf{j}) \]
\[ \mathbf{b} + (-\mathbf{c}) = (6 + 1) \mathbf{i} + (2 – 7) \mathbf{j} \]
\[ \mathbf{b} + (-\mathbf{c}) = 7 \mathbf{i} – 5 \mathbf{j} \]
Kahoo, sephetho sa \(\mathbf{b} + (-\mathbf{c})\) ke:
\[ 7 \ lipalobf{i} - 5 \ lipalo{j} \]
Mohlala oa 3:
Ho na le vector \(\mathbf{d} = a \mathbf{i} + b \mathbf{j}\), moo a le b e leng linomoro tsa sebele. Haeba \(\mathbf{d} + \mathbf{e} = \mathbf{0}\), khetha vector \(\mathbf{e}\).
Puisano:
Hoa tsebahala:
\[ \mathbf{d} = a \mathbf{i} + b \mathbf{j} \]
\[ \mathbf{d} + \mathbf{e} = \mathbf{0} \]
Ho fumana \(\mathbf{e}\), re ka ngola:
\[ \mathbf{e} = -\mathbf{d} \]
Kahoo, vekthara \(\mathbf{e}\) ke vekthara e mpe ea \(\mathbf{d}\):
\[ \mathbf{e} = -\mathbf{d} = -a \mathbf{i} – b \mathbf{j} \]
Mohlala oa 4:
Ho fanoe ka vector \(\mathbf{f} = 5 \mathbf{i} + k \mathbf{j}\). Hoa tsebahala hore vector e mpe ea \(\mathbf{f}\) ke \(-5 \mathbf{i} - 8 \mathbf{j}\). Fumana boleng ba k.
Puisano:
Hoa tsebahala:
\[ \mathbf{f} = 5 \mathbf{i} + k \mathbf{j} \]
\[ -\mathbf{f} = -5 \mathbf{i} – 8 \mathbf{j} \]
Ho tsoa kamanong ena, re ka haha di-equation tsa dikarolo bakeng sa \(\mathbf{f}\) le \(-\mathbf{f}\). Ka karolo, vekthara \(\mathbf{f}\) le vekthara ya yona e mpe di lokela ho ba le kamano e tshwanang ya boemo le matshwao a fapaneng. Kahoo:
Bakeng sa likarolo \( \mathbf{i} \):
\[ -5 = -5 \]
Sena ke 'nete ka bohona.
Bakeng sa karolo \( \mathbf{j} \):
\[ -k = -8 \]
k = 8
Kahoo, boleng ba \( k \) ke 8.
Qetello
Ho utloisisa mohopolo oa vector e mpe, kapa vector e fapaneng, ho bohlokoa ho ithuteng li-vector. Vector e fapaneng ke vector e fapaneng ka lehlakoreng le leng le vector ea pele empa e na le boholo bo tšoanang. Ts'ebetsong ea vector, ho lemoha le ho sebelisa li-vector tse mpe ho ka thusa haholo ho nolofatsa mathata a mangata, joalo ka ho eketsa kapa ho tlosa li-vector. Ka ho itloaetsa le ho utloisisa litšobotsi tsa motheo tsa li-vector, ho utloisisa mohopolo ona ho tla ba bonolo haholoanyane.
Re tšepa hore mehlala ea lipotso le puisano tse hlahisitsoeng sehloohong sena li tla u thusa ho utloisisa ka botebo li-vector tse mpe, kapa li-vector tse hanyetsanang. Tsoela pele ho itlhakisa le ho hlahloba lipotso tse ling ho ba le boiphihlelo bo eketsehileng boitsebisong bona!
