Nā nīnau hoʻohālike a me ke kūkākūkā ʻana o ka Vector Subtraction
Pendahuluan
I ka makemakika a me ke kino, he manaʻo nui nā vectors i hoʻohana ʻia e wehewehe i nā hanana kūlohelohe a me nā ʻenekinia. ʻO ka vector kahi nui i loaʻa ka nui a me ke kuhikuhi. ʻO kekahi mau hiʻohiʻona koʻikoʻi o nā vectors ka neʻe ʻana, ka wikiwiki, ka wikiwiki, a me ka ikaika. Ma kēia ʻatikala, e kūkākūkā mākou i ka unuhi vector, ʻoiai ua hoʻokūpaʻa pinepine ʻia kēia kumuhana ma ke ʻano o ka hui pū ʻana o nā vector.
He hana koʻikoʻi ka unuhi vector i ka nānā ʻana i ka vector. No ka hohonu ʻana i kēia manaʻo, e nānā kākou i kekahi mau pilikia hoʻohālike a me nā kūkākūkā e pili ana i ka unuhi vector.
Ka Hoʻemi Vector
Ua wehewehe ʻia ka vector subtraction {\displaystyle \mathbf{A} – \mathbf{B}} ʻo ia ka hana o ka hoʻohui ʻana i ka vector {\displaystyle \mathbf{A}} me ka vector {\displaystyle -\mathbf{B}}, kahi {\displaystyle -\mathbf{B}} he vector me ka nui like me {\displaystyle \mathbf{B}} akā me ke kuhikuhi ʻē aʻe. Ma ke ʻano makemakika, hiki ke kākau ʻia kēia penei:
{\displaystyle \mathbf{A} – \mathbf{B} = \mathbf{A} + (-\mathbf{B})}
Nā Nīnau Laʻana a me ke Kūkākūkā
Nīnau 1: Ke unuhi ʻana i nā Vectors ʻElua-Dimensional
Manaʻo ʻia he ʻelua mau vectors ma nā hoʻonohonoho Cartesian:
{\displaystyle \mathbf{A} = (4, 3)} a me {\displaystyle \mathbf{B} = (1, 2)}. E helu i {\displaystyle \mathbf{A} – \mathbf{B}}.
Kūkākūkā:
ʻO ka hana mua, ʻo ia ke ʻimi i ka vector maikaʻi ʻole o {\displaystyle \mathbf{B}}, ʻo ia hoʻi:
{\displaystyle -\mathbf{B} = (-1, -2)}
A laila, e hoʻohui i ka vector {\displaystyle \mathbf{A}} me {\displaystyle -\mathbf{B}}:
{\displaystyle \mathbf{A} – \mathbf{B} = (4, 3) + (-1, -2)}
E hana i ka hoʻohui vector ma ka hoʻohui ʻana i kēlā me kēia ʻāpana x a me y:
{\displaystyle \mathbf{A} – \mathbf{B} = (4 + (-1), 3 + (-2))}
{\displaystyle \mathbf{A} – \mathbf{B} = (3, 1)}
No laila, ʻo ka hopena o ka unuhi ʻana i nā vectors {\displaystyle \mathbf{A} – \mathbf{B}} ʻo ia ka vector (3, 1).
Nīnau 2: Ke unuhi ʻana i nā Vectors ʻEkolu-Dimensional
Hāʻawi ʻia i ʻelua mau vectors ma nā hoʻonohonoho ʻekolu-dimensional:
{\displaystyle \mathbf{P} = (2, -4, 6)} a me {\displaystyle \mathbf{Q} = (-3, 5, 7)}. E helu i {\displaystyle \mathbf{P} – \mathbf{Q}}.
Kūkākūkā:
ʻO ka hana mua, ʻo ia ke ʻimi i ka vector maikaʻi ʻole o {\displaystyle \mathbf{Q}}:
{\displaystyle -\mathbf{Q} = (3, -5, -7)}
A laila, e hoʻohui i ka vector {\displaystyle \mathbf{P}} me {\displaystyle -\mathbf{Q}}:
{\displaystyle \mathbf{P} – \mathbf{Q} = (2, -4, 6) + (3, -5, -7)}
E hana i ka hoʻohui vector ma ka hoʻohui ʻana i kēlā me kēia ʻāpana x, y, a me z:
{\displaystyle \mathbf{P} – \mathbf{Q} = (2 + 3, -4 + (-5), 6 + (-7))}
{\displaystyle \mathbf{P} – \mathbf{Q} = (5, -9, -1)}
No laila, ʻo ka hopena o ka unuhi ʻana i nā vectors {\displaystyle \mathbf{P} – \mathbf{Q}} ʻo ia ka vector (5, -9, -1).
Nīnau 3: Ka Hoʻemi Vector ma ka Papahele Paʻakikī
Manaʻo ʻia he ʻelua mau vectors i hōʻike ʻia e nā helu paʻakikī:
{\displaystyle \mathbf{M} = 3 + 4i} a me {\displaystyle \mathbf{N} = 1 + 2i}. E helu i {\displaystyle \mathbf{M} – \mathbf{N}}.
Kūkākūkā:
ʻO ka hana mua, ʻo ia ke ʻimi i ka vector maikaʻi ʻole o {\displaystyle \mathbf{N}}:
{\displaystyle -\mathbf{N} = -1 – 2i}
A laila, e hoʻohui i ka vector {\displaystyle \mathbf{M}} me {\displaystyle -\mathbf{N}}:
{\displaystyle \mathbf{M} – \mathbf{N} = (3 + 4i) + (-1 – 2i)}
E hana i ka hoʻohui vector ma ka hoʻohui ʻana i kēlā me kēia ʻāpana maoli a me ka ʻāpana hoʻokalakupua:
{\displaystyle \mathbf{M} – \mathbf{N} = (3 + (-1)) + (4i + (-2i))}
{\displaystyle \mathbf{M} – \mathbf{N} = 2 + 2i}
No laila, ʻo ka hopena o ka unuhi ʻana i nā vectors {\displaystyle \mathbf{M} – \mathbf{N}} ʻo ia ka helu paʻakikī 2 + 2i.
Nīnau 4: Ka Hoʻemi Vector ma ka ʻŌnaehana Hoʻonohonoho Polar
Manaʻo ʻia he ʻelua mau vectors i nā hoʻonohonoho polar:
He 5 ka nui o {\displaystyle \mathbf{U}} a he 30° ke kihi,
a he 3 ka nui o {\displaystyle \mathbf{V}} a he 150° ke kihi.
E helu i {\displaystyle \mathbf{U} – \mathbf{V}}.
Kūkākūkā:
ʻO ka hana mua, ʻo ia ka hoʻololi ʻana i nā vectors {\displaystyle \mathbf{U}} a me {\displaystyle \mathbf{V}} i nā hoʻonohonoho Cartesian.
No {\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}
No laila, ʻo {\displaystyle \mathbf{U}} ma Cartesian ʻo (4.33, 2.5).
No {\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}
No laila, ʻo {\displaystyle \mathbf{V}} ma Cartesian ʻo (-2.598, 1.5).
ʻO ka hana aʻe, e helu i ka unuhi vector ma Cartesian:
{\displaystyle \mathbf{U} – \mathbf{V} = (4.33, 2.5) – (-2.598, 1.5)}
ʻO ia hoʻi ke hoʻohui ʻana i ka maikaʻi ʻole o ka vector:
{\displaystyle \mathbf{U} – \mathbf{V} = (4.33 + 2.598, 2.5 – 1.5)}
{\displaystyle \mathbf{U} – \mathbf{V} = (6.928, 1)}
No laila, ʻo ka hopena o ka unuhi ʻana i ka vector {\displaystyle \mathbf{U} – \mathbf{V}} ma nā hoʻonohonoho Cartesian ʻo ia (6.928, 1).
Ka hopena
He hana makemakika koʻikoʻi ka hoʻemi ʻana o ka vector ma nā ʻano he nui e hoʻohana ana i ka nānā ʻana o ka vector. Inā paha ma nā ʻōnaehana hoʻonohonoho ʻelua-dimensional, ʻekolu-dimensional, paʻakikī, a i ʻole polar, ua like ke kumu nui: e hoʻohui i hoʻokahi vector i ka maikaʻi ʻole o kekahi. Hōʻike nā hiʻohiʻona ma luna i nā ʻano like ʻole e hoʻopili ai i kēia hana ma nā ʻano like ʻole, e kōkua ana iā mākou e hoʻomaopopo hohonu a me ka hana maoli.
