Imibuzo Yezibonelo kanye Nengxoxo Yokususa Ama-Vector
I-Pendahuluan
Kumathematika nakufiziksi, ama-vector angumqondo oyisisekelo osetshenziswa ukuchaza izinto eziningi zemvelo nezobunjiniyela. I-vector inani elinobukhulu kanye nesiqondiso. Ezinye izibonelo ezibalulekile zama-vector ukufuduka, ijubane, ukusheshisa, kanye namandla. Kulesi sihloko, sizoxoxa ngokukhipha ama-vector, yize lesi sihloko sivame ukugcizelelwa kumongo wokuhlanganiswa kwama-vector.
Ukususa amavektha kuwumsebenzi oyisisekelo obalulekile ekuhlaziyweni kwamavektha. Ukuze singene shí kulo mqondo, ake sibukeze ezinye zezibonelo zezinkinga kanye nezingxoxo ezihlobene nokususa amavektha.
Ukususa Amavektha
Ukususa i-vector {\displaystyle \mathbf{A} – \mathbf{B}} kuchazwa njengokusebenza kokwengeza i-vector {\displaystyle \mathbf{A}} ne-vector {\displaystyle -\mathbf{B}}, lapho i-{\displaystyle -\mathbf{B}} iyi-vector enobukhulu obufanayo ne-{\displaystyle \mathbf{B}} kodwa enohlangothi oluphambene. Ngokwezibalo, lokhu kungabhalwa kanje:
{\displaystyle \mathbf{A} – \mathbf{B} = \mathbf{A} + (-\mathbf{B})}
Imibuzo Eyisibonelo Nengxoxo
Umbuzo 1: Ukususa Amavektha Anezinhlangothi Ezimbili
Ake sithi kunezivektha ezimbili kuzixhumanisi zeCartesian:
{\displaystyle \mathbf{A} = (4, 3)} kanye {\displaystyle \mathbf{B} = (1, 2)}. Bala {\displaystyle \mathbf{A} – \mathbf{B}}.
Ingxoxo:
Isinyathelo sokuqala ukuthola ivektha engemihle ye- {\displaystyle \mathbf{B}}, okungukuthi:
{\displaystyle -\mathbf{B} = (-1, -2)}
Okulandelayo, engeza i-vector {\displaystyle \mathbf{A}} nge- {\displaystyle -\mathbf{B}}:
{\displaystyle \mathbf{A} – \mathbf{B} = (4, 3) + (-1, -2)}
Yenza ukwengeza i-vector ngokungeza ingxenye ngayinye ye-x ne-y:
{\displaystyle \mathbf{A} – \mathbf{B} = (4 + (-1), 3 + (-2))}
{\displaystyle \mathbf{A} – \mathbf{B} = (3, 1)}
Ngakho-ke, umphumela wokukhipha amavekhtha {\displaystyle \mathbf{A} – \mathbf{B}} yivekhtha (3, 1).
Umbuzo 2: Ukususa Amavektha Anezinhlangothi Ezintathu
Kunikezwe amavekhtha amabili kuzixhumanisi ezinezinhlangothi ezintathu:
{\displaystyle \mathbf{P} = (2, -4, 6)} kanye {\displaystyle \mathbf{Q} = (-3, 5, 7)}. Bala {\displaystyle \mathbf{P} – \mathbf{Q}}.
Ingxoxo:
Isinyathelo sokuqala ukuthola ivektha engemihle ye- {\displaystyle \mathbf{Q}}:
{\displaystyle -\mathbf{Q} = (3, -5, -7)}
Okulandelayo, engeza i-vector {\displaystyle \mathbf{P}} nge- {\displaystyle -\mathbf{Q}}:
{\displaystyle \mathbf{P} – \mathbf{Q} = (2, -4, 6) + (3, -5, -7)}
Yenza ukwengeza i-vector ngokwengeza ingxenye ngayinye ye-x, y, kanye ne-z:
{\displaystyle \mathbf{P} – \mathbf{Q} = (2 + 3, -4 + (-5), 6 + (-7))}
{\displaystyle \mathbf{P} – \mathbf{Q} = (5, -9, -1)}
Ngakho-ke, umphumela wokukhipha amavekhtha {\displaystyle \mathbf{P} – \mathbf{Q}} yivekhtha (5, -9, -1).
Umbuzo 3: Ukususwa kweVektha eNdizeni Eyinkimbinkimbi
Ake sithi kuneziveji ezimbili ezimelelwe izinombolo eziyinkimbinkimbi:
{\displaystyle \mathbf{M} = 3 + 4i} kanye {\displaystyle \mathbf{N} = 1 + 2i}. Bala {\displaystyle \mathbf{M} – \mathbf{N}}.
Ingxoxo:
Isinyathelo sokuqala ukuthola ivektha engemihle ye- {\displaystyle \mathbf{N}}:
{\displaystyle -\mathbf{N} = -1 – 2i}
Okulandelayo, engeza i-vector {\displaystyle \mathbf{M}} nge- {\displaystyle -\mathbf{N}}:
{\displaystyle \mathbf{M} – \mathbf{N} = (3 + 4i) + (-1 – 2i)}
Yenza ukwengeza i-vector ngokungeza ingxenye ngayinye yangempela neyinganekwane:
{\displaystyle \mathbf{M} – \mathbf{N} = (3 + (-1)) + (4i + (-2i))}
{\displaystyle \mathbf{M} – \mathbf{N} = 2 + 2i}
Ngakho-ke, umphumela wokukhipha amavekhtha {\displaystyle \mathbf{M} – \mathbf{N}} inombolo eyinkimbinkimbi 2 + 2i.
Umbuzo 4: Ukususwa kweVektha ohlelweni lwe-Polar Coordinate
Ake sithi kunezivektha ezimbili ku-polar coordinates:
I-{\displaystyle \mathbf{U}} inobukhulu obungu-5 kanye ne-engeli engu-30°,
futhi i-{\displaystyle \mathbf{V}} inobukhulu obungu-3 kanye ne-engeli engu-150°.
Bala {\displaystyle \mathbf{U} – \mathbf{V}}.
Ingxoxo:
Isinyathelo sokuqala ukuguqula amavekhtha {\displaystyle \mathbf{U}} kanye ne-{\displaystyle \mathbf{V}} abe ama-coordinates e-Cartesian.
Okwe-{\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}
Ngakho-ke {\displaystyle \mathbf{U}} ku-Cartesian ingu-(4.33, 2.5).
Okwe-{\displaystyle \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}
Ngakho-ke i- {\displaystyle \mathbf{V}} ku-Cartesian ingu-(-2.598, 1.5).
Isinyathelo esilandelayo, bala ukususwa kwevektha ku-Cartesian:
{\displaystyle \mathbf{U} – \mathbf{V} = (4.33, 2.5) – (-2.598, 1.5)}
Okusho ukuthi ngokungeza i-negative ye-vector:
{\displaystyle \mathbf{U} – \mathbf{V} = (4.33 + 2.598, 2.5 – 1.5)}
{\displaystyle \mathbf{U} – \mathbf{V} = (6.928, 1)}
Ngakho-ke, umphumela wokukhipha i-vector {\displaystyle \mathbf{U} – \mathbf{V}} kuma-coordinates e-Cartesian ngu-(6.928, 1).
Isiphetho
Ukususa amavektha kuyindlela ebalulekile yezibalo emikhakheni eminingi esebenzisa ukuhlaziywa kwamavektha. Kungakhathaliseki ukuthi kusezinhlelweni zokuxhumanisa ezinezinhlangothi ezimbili, ezinezinhlangothi ezintathu, eziyinkimbinkimbi, noma ezinezinhlangothi eziphansi, isimiso esiyisisekelo sihlala sifana: ukwengeza ivektha eyodwa kokungafaneleki kwenye. Izibonelo ezingenhla zibonisa izindlela ezahlukene zokusebenzisa lo msebenzi ezimweni ezahlukene, okusisiza siqonde umqondo ngokujulile nangokusebenzayo.
