Mehlala ea Lipotso Tse Buisanang ka Li-vector tsa Mahlakore a Mabeli Tsamaisong ea Khokahano
Vekthara ke bongata bo nang le boholo le tataiso. Livekthara li sebelisoa khafetsa lihloohong tse fapaneng tsa lipalo le fisiks ho emela liketsahalo tse fapaneng. Sehloohong sena, re tla tšohla mehlala ea livekthara tse nang le mahlakore a mabeli tsamaisong ea coordinate.
Mehopolo ea Motheo ea Li-vector ho Litsamaiso tse Kopanetsoeng
Vekthara tsamaisong ea likhokahano tsa mahlakore a mabeli e ka emeloa e le \(\vec{A} = (a_1, a_2)\), moo \(a_1\) e leng karolo ea x ea vekthara 'me \(a_2\) e le karolo ea y ea vekthara. Vekthara e ka aroloa likarolo tse peli, e leng karolo ea x le karolo ea y.
Ho eketsa le ho tlosa vector
Ho eketsoa ha divekthara tse pedi \(\vec{A} = (a_1, a_2)\) le \(\vec{B} = (b_1, b_2)\) ke:
\[
\vec{A} + \vec{B} = (a_1 + b_1, a_2 + b_2)
\]
Leha phokotso e le:
\[
\vec{A} – \vec{B} = (a_1 – b_1, a_2 – b_2)
\]
Ho Atisa ha Scalar
Haeba \(\vec{A} = (a_1, a_2)\) le \(k\) e le scalar, joale \(k\vec{A}\) ke:
\[
k\vec{A} = (k \cdot a_1, k \cdot a_2)
\]
Boholo ba Vekthara
Boholo kapa bolelele ba vekthara \(\vec{A} = (a_1, a_2)\) ke:
\[
|\vec{A}| = \sqrt{a_1^2 + a_2^2}
\]
Vektheri ea Yuniti
Vekthara ea yuniti ke vekthara e nang le bolelele ba yuniti e le 'ngoe. Vekthara ea yuniti ea \(\vec{A} = (a_1, a_2)\) ke:
\[
\hat{A} = \frac{\vec{A}}{|\vec{A}|} = \left( \frac{a_1}{\sqrt{a_1^2 + a_2^2}}, \frac{a_2}{\sqrt{a_1^2 + a_2^2}} \right)
\]
Lipotso tsa Mehlala le Puisano
Potso ea 1: Ho eketsa le ho tlosa li-vector
Ho fanoe ka livekthara tse peli ka tsela e latelang: \(\vec{A} = (3, 4)\) le \(\vec{B} = (1, 2)\). Fumana sehlahisoa sa \(\vec{A} + \vec{B}\) le \(\vec{A} – \vec{B}\).
Puisano:
\[
\vec{A} + \vec{B} = (3 + 1, 4 + 2) = (4, 6)
\]
\[
\vec{A} – \vec{B} = (3 – 1, 4 – 2) = (2, 2)
\]
Potso ea 2: Katiso ea Scalar
Ha ho fanoe ka vekthara \(\vec{C} = (2, -3)\), bala \(3\vec{C}\) le \(-2\vec{C}\).
Puisano:
\[
3\vec{C} = 3 \cdot (2, -3) = (6, -9)
\]
\[
-2\vec{C} = -2 \cdot (2, -3) = (-4, 6)
\]
Potso ea 3: Boholo ba Vektara
Bala boholo ba vekthara \(\vec{D} = (5, 12)\).
Puisano:
\[
|\vec{D}| = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]
Potso ea 4: Li-vector tsa Yuniti
Fumana vekthara ea yuniti ea vekthara \(\vec{E} = (4, 3)\).
Puisano:
\[
|\vec{E}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \ sqrt{25} = 5
\]
\[
\hat{E} = \frac{\vec{E}}{|\vec{E}|} = \left( \frac{4}{5}, \frac{3}{5} \right)
\]
Potso ea 5: Sebaka le Sebaka sa Livekthara
Lintlha tse peli tse ka har'a sefofane sa coordinate sa mahlakore a mabeli ke P(2, 3) le Q(5, 7). Fumana vector ea boemo ho tloha ntlheng ea P ho ea ntlheng ea Q le sebaka se pakeng tsa tsona.
Puisano:
Vekthara ea boemo ho tloha ho P ho ea ho Q ke:
\[
\vec{PQ} = \vec{Q} – \vec{P} = (5 – 2, 7 – 3) = (3, 4)
\]
Sebaka se pakeng tsa lintlha P le Q ke:
\[
|\vec{PQ}| = \ sqrt{(3)^2 + (4)^2} = \ sqrt{9 + 16} = \ sqrt{25} = 5
\]
Potso ea 6: Sephetho sa Sehlahisoa sa Dot
Haeba \(\vec{F} = (-3, 4)\) le \(\vec{G} = (2, 1)\), bala sehlahiswa sa matheba sa \(\vec{F} \cdot \vec{G}\).
Puisano:
Sehlahisoa sa matheba sa li-vector tse peli ke:
\[
\vec{F} \cdot \vec{G} = (-3) \cdot 2 + 4 \cdot 1 = -6 + 4 = -2
\]
Potso ea 7: Sekhutlo Pakeng tsa Livektara tse peli
Haeba \(\vec{H} = (7, -4)\) le \(\vec{I} = (3, 0)\), fumana sekhutlo pakeng tsa divekthara tse pedi.
Puisano:
Ho fumana sekhutlo pakeng tsa li-vector tse peli, re sebelisa foromo:
\[
\cos \theta = \frac{\vec{H} \cdot \vec{I}}{|\vec{H}| |\vec{I}|}
\]
Taba ea pele, bala sehlahisoa sa matheba \(\vec{H} \cdot \vec{I}\):
\[
\vec{H} \cdot \vec{I} = 7 \cdot 3 + (-4) \cdot 0 = 21 + 0 = 21
\]
Ebe, bala boholo ba \(\vec{H}\) le \(\vec{I}\):
\[
|\vec{H}| = \sqrt{7^2 + (-4)^2} = \sqrt{49 + 16} = \sqrt{65}
\]
\[
|\vec{I}| = \sqrt{3^2 + 0^2} = \sqrt{9} = 3
\]
Kenya boleng bona ka foromong:
\[
\cos \theta = \frac{21}{\sqrt{65} \cdot 3} = \frac{21}{3\sqrt{65}} = \frac{7}{\sqrt{65}}
\]
Kahoo, \(\theta = \cos^{-1} \left( \frac{7}{\sqrt{65}} \right) \).
Potso ea 8: Ponelopele ea Vektara
Bakeng sa divekthara \(\vec{J} = (2, 1)\) le \(\vec{K} = (-1, 3)\), bala ponelopele ya \(\vec{J}\) ho \(\vec{K}\).
Puisano:
Ponelopele ea \(\vec{J}\) ho \(\vec{K}\) ke:
\[
\mongolo{proj}_{\vec{K}} \vec{J} = \left( \frac{\vec{J} \cdot \vec{K}}{|\vec{K}|^2} \ka ho le letona) \vec{K}
\]
Taba ea pele, bala sehlahisoa sa matheba \(\vec{J} \cdot \vec{K}\):
\[
\vec{J} \cdot \vec{K} = 2 \cdot (-1) + 1 \cdot 3 = -2 + 3 = 1
\]
Ebe, boholo ba \(\vec{K}\):
\[
|\vec{K}| = \sqrt{(-1)^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}
\]
E le hore,:
\[
|\vec{K}|^2 = 10
\]
Kenya foromong:
\[
\text{proj}_{\vec{K}} \vec{J} = \left( \frac{1}{10} \right) \vec{K} = \left( \frac{1}{10} \right) (-1, 3) = \left( -\frac{1}{10}, \frac{3}{10} \right)
\]
Tsena ke mehlala ea mathata le lipuisano tse amanang le li-vector tsa mahlakore a mabeli tsamaisong ea likhokahano. Kutloisiso e ntle ea li-vector e ka ba molemo lits'ebetsong tse ngata tsa lipalo, fisiks le boenjiniere. Ho itlhakisa ka mehlala e fapaneng ho ka tebisa kutloisiso ea hau ea mohopolo ona, ho o lumella ho sebelisoa ka katleho maemong a fapaneng.
