Mehlala ea Lipotso le Puisano ea ho Ntša Vector
Pendahuluan
Lipalong le fisiks, li-vector ke mohopolo oa motheo o sebelisoang ho hlalosa liketsahalo tse ngata tsa tlhaho le tsa boenjiniere. Vector ke bongata bo nang le boholo le tataiso. Mehlala e meng ea bohlokoa ea li-vector ke ho falla, lebelo, ho potlaka le matla. Sehloohong sena, re tla buisana ka ho ntša li-vector, leha taba ena e atisa ho hatisoa moelelong oa motsoako oa li-vector.
Ho ntsha divektara ke tshebetso ya motheo e bohlokwa haholo tlhahlobong ya divektara. Ho teba ka botebo mohopolong ona, ha re hlahlobeng mehlala e meng ya mathata le dipuisano tse amanang le ho ntsha divektara.
Ho Ntša Vector
Ho tlosa vekthara {\displaystyle \mathbf{A} – \mathbf{B}} ho hlaloswa e le tshebetso ya ho eketsa vekthara {\displaystyle \mathbf{A}} le vekthara {\displaystyle -\mathbf{B}}, moo {\displaystyle -\mathbf{B}} e leng vekthara e nang le boholo bo lekanang le {\displaystyle \mathbf{B}} empa e le ka lehlakoreng le fapaneng. Ka dipalo, sena se ka ngolwa tjena:
{\ mokhoa oa ho bonts'a \mathbf{A} – \mathbf{B} = \mathbf{A} + (-\mathbf{B})}
Lipotso tsa Mehlala le Puisano
Potso ea 1: Ho Ntša Livekthara tsa Mahlakore a Mabeli
A re re ho na le li-vector tse peli ho li-coordinate tsa Cartesian:
{\displaystyle \mathbf{A} = (4, 3)} le {\displaystyle \mathbf{B} = (1, 2)}. Bala {\displaystyle \mathbf{A} – \mathbf{B}}.
Puisano:
Mohato oa pele ke ho fumana vekthara e mpe ea {\displaystyle \mathbf{B}}, e leng:
{\displaystyle -\mathbf{B} = (-1, -2)}
Ka mor'a moo, eketsa vektha {\displaystyle \mathbf{A}} ka {\displaystyle -\mathbf{B}}:
{\ mokhoa oa ho bonts'a \mathbf{A} – \mathbf{B} = (4, 3) + (-1, -2)}
Etsa tlatsetso ea vector ka ho eketsa karolo e 'ngoe le e 'ngoe ea x le y:
{\ mokhoa oa ho bonts'a \mathbf{A} – \mathbf{B} = (4 + (-1), 3 + (-2))}
{\ mokhoa oa ho bonts'a \mathbf{A} – \mathbf{B} = (3, 1)}
Kahoo, sephetho sa ho tlosa divekthara {\displaystyle \mathbf{A} – \mathbf{B}} ke vekthara (3, 1).
Potso ea 2: Ho Ntša Livekthara tsa Mahlakore a Mararo
Ho fanoe ka li-vector tse peli ka li-coordinate tsa mahlakore a mararo:
{\displaystyle \mathbf{P} = (2, -4, 6)} le {\displaystyle \mathbf{Q} = (-3, 5, 7)}. Bala {\displaystyle \mathbf{P} – \mathbf{Q}}.
Puisano:
Mohato oa pele ke ho fumana vekthara e mpe ea {\displaystyle \mathbf{Q}}:
{\displaystyle -\mathbf{Q} = (3, -5, -7)}
Ka mor'a moo, eketsa vektha {\displaystyle \mathbf{P}} ka {\displaystyle -\mathbf{Q}}:
{\ mokhoa oa ho bonts'a \mathbf{P} – \mathbf{Q} = (2, -4, 6) + (3, -5, -7)}
Etsa tlatsetso ea vector ka ho eketsa karolo e 'ngoe le e 'ngoe ea x, y, le z:
{\ mokhoa oa ho bonts'a \mathbf{P} – \mathbf{Q} = (2 + 3, -4 + (-5), 6 + (-7))}
{\ mokhoa oa ho bonts'a \mathbf{P} – \mathbf{Q} = (5, -9, -1)}
Kahoo, sephetho sa ho tlosa divekthara {\displaystyle \mathbf{P} – \mathbf{Q}} ke vekthara (5, -9, -1).
Potso ea 3: Ho Ntša Vector ka Sefofaneng se Rarahaneng
A re re ho na le divekthara tse pedi tse emetsweng ke dinomoro tse rarahaneng:
{\displaystyle \mathbf{M} = 3 + 4i} le {\displaystyle \mathbf{N} = 1 + 2i}. Bala {\displaystyle \mathbf{M} – \mathbf{N}}.
Puisano:
Mohato oa pele ke ho fumana vekthara e mpe ea {\displaystyle \mathbf{N}}:
{\ mokhoa oa ho bonts'a -\mathbf{N} = -1 – 2i}
Ka mor'a moo, eketsa vektha {\displaystyle \mathbf{M}} ka {\displaystyle -\mathbf{N}}:
{\ mokhoa oa ho bonts'a \mathbf{M} – \mathbf{N} = (3 + 4i) + (-1 – 2i)}
Etsa tlatsetso ea vector ka ho eketsa karolo e 'ngoe le e 'ngoe ea 'nete le ea boiqapelo:
{\ mokhoa oa ho bonts'a \mathbf{M} – \mathbf{N} = (3 + (-1)) + (4i + (-2i))}
{\ mokhoa oa ho bonts'a \mathbf{M} – \mathbf{N} = 2 + 2i}
Kahoo, sephetho sa ho tlosa divektha {\displaystyle \mathbf{M} – \mathbf{N}} ke nomoro e rarahaneng ya 2 + 2i.
Potso ea 4: Ho Ntša Vector ho Polar Coordinate System
A re re ho na le li-vector tse peli likhokahanong tsa polar:
{\displaystyle \mathbf{U}} e na le boholo ba 5 le sekhutlo sa 30°,
mme {\displaystyle \mathbf{V}} e na le boholo ba 3 le sekhutlo sa 150°.
Bala {\displaystyle \mathbf{U} – \mathbf{V}}.
Puisano:
Mohato oa pele ke ho fetolela li-vector {\displaystyle \mathbf{U}} le {\displaystyle \mathbf{V}} ho li-coordinate tsa Cartesian.
Bakeng sa {\ displaystyle \mathbf{U}}:
{\displaystyle U_x = 5 \cos(30^\circ) = 5 \left(\frac{\sqrt{3}}{2}\right) = 5 \cdot 0.866 = 4.33}
{\displaystyle U_y = 5 \sin(30^\circ) = 5 \left(\frac{1}{2}\right) = 5 \cdot 0.5 = 2.5}
Kahoo {\displaystyle \mathbf{U}} ho Cartesian ke (4.33, 2.5).
Bakeng sa {\playstyle \mathbf{V}}:
{\displaystyle V_x = 3 \cos(150^\circ) = 3 \left(\frac{-\sqrt{3}}{2}\right) = 3 \cdot (-0.866) = -2.598}
{\displaystyle V_y = 3 \sin(150^\circ) = 3 \left(\frac{1}{2}\right) = 3 \cdot 0.5 = 1.5}
Kahoo {\displaystyle \mathbf{V}} ho Cartesian ke (-2.598, 1.5).
Mohato o latelang, bala ho ntsha vector ka Cartesian:
{\displaystyle \mathbf{U} – \mathbf{V} = (4.33, 2.5) – (-2.598, 1.5)}
Ho bolelang ka ho eketsa negative ea vector:
{\displaystyle \mathbf{U} – \mathbf{V} = (4.33 + 2.598, 2.5 – 1.5)}
{\displaystyle \mathbf{U} – \mathbf{V} = (6.928, 1)}
Kahoo, sephetho sa ho tlosa vektha {\displaystyle \mathbf{U} – \mathbf{V}} ho di-coordinate tsa Cartesian ke (6.928, 1).
Qetello
Ho tlosa divektara ke tshebetso ya bohlokwa ya dipalo masimong a mangata a sebedisang tlhahlobo ya divektara. Ebang ke ditsamaisong tsa di-coordinate tsa mahlakore a mabedi, mahlakore a mararo, tse rarahaneng, kapa tsa polar, molao-motheo wa motheo o dula o tshwana: ho eketsa vektara e le nngwe ho negative ya e nngwe. Mehlala e ka hodimo e bontsha ditsela tse fapaneng tsa ho sebedisa tshebetso ena maemong a fapaneng, e re thusang ho utlwisisa mohopolo ka botebo le ka tsela e sebetsang.
