Laʻana o nā pilikia e kūkākūkā ana i nā Vectors Kūlana
He manaʻo nui nā vectors i ka makemakika a me ke kino, e hōʻike ana i nā nui me ke kuhikuhi a me ka nui. Ma nā ʻano noi like ʻole, hoʻohana pinepine ʻia nā vectors e wehewehe i ke kūlana, ka wikiwiki, ka ikaika, a me nā ʻano ʻē aʻe he nui. Ma waena o nā ʻano vectors like ʻole, he kuleana koʻikoʻi nā vectors kūlana i ke kaha palapala ʻana i kahi o kahi kiko ma ka lewa.
Ka Wehewehena o ke Vector Kūlana
ʻO ka vector kūlana kahi vector e wehewehe ana i kahi o kahi kiko e pili ana i ke kumu i loko o kahi ʻōnaehana hoʻonohonoho. Ma keʻano laulā, ua kākau ʻia kahi vector kūlana ma ke ʻano hoʻonohonoho Cartesian penei:
\[ \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \]
Maanei, ʻo \(\mathbf{r}\) ka vector kūlana, ʻo \(x\), \(y\), a me \(z\) kona mau ʻāpana ma nā axes \(x\), \(y\), a me \(z\), kēlā me kēia, ʻoiai ʻo \(\mathbf{i}\), \(\mathbf{j}\), a me \(\mathbf{k}\) he mau vectors unit e kūlike ana i nā axes coordinate, kēlā me kēia. Ma ka hakahaka ʻelua-dimensional, ʻaʻole loaʻa ka ʻāpana \(z\), no laila lilo ka vector kūlana:
\[ \mathbf{r} = x\mathbf{i} + y\mathbf{j} \]
Nā noi Vector Kūlana
Eia kekahi laʻana, i ka physics, he kuleana koʻikoʻi ko nā vectors kūlana i ka wehewehe ʻana i ka neʻe ʻana o nā mea. Hiki ke hōʻike ʻia ke kūlana o kahi mea e pili ana i ke kumu (kiko kuhikuhi) e kahi vector kūlana. Eia kekahi, i ka ʻenekinia mechanical, pili pinepine ka helu ʻana o nā ikaika a me nā manawa i ka hoʻohana ʻana i nā vectors kūlana.
Nā nīnau hoʻohālike a me ke kūkākūkā ʻana o nā Vectors Kūlana
Nīnau 1
Manaʻo ʻia he ʻelua mau kiko ma ka lewa 3D, ʻo ke kiko A me nā hoʻonohonoho \( (1, 2, 3) \) a me ke kiko B me nā hoʻonohonoho \( (4, 0, -2) \). E hoʻoholo i nā vectors kūlana o nā kiko A a me B. Eia kekahi, e helu i ka vector e hoʻopili ana i ke kiko A i ke kiko B.
Kūkākūkā:
Vector kūlana no ke kiko A:
\[ \mathbf{r_A} = 1\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \]
Vector kūlana no ke kiko B:
\[ \mathbf{r_B} = 4\mathbf{i} + 0\mathbf{j} – 2\mathbf{k} \]
ʻO ka mea aʻe, no ka loaʻa ʻana o ka vector e hoʻopili ana i ke kiko A i ke kiko B (i kapa ʻia ʻo \(\mathbf{AB}\)), pono mākou e unuhi i ka vector kūlana o A mai ka vector kūlana o B:
\[ \mathbf{AB} = \mathbf{r_B} – \mathbf{r_A} \]
No laila, ke pani nei i nā vectors kūlana ʻelua ma luna:
\[ \mathbf{AB} = (4\mathbf{i} + 0\mathbf{j} – 2\mathbf{k}) – (1\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}) \]
\[ \mathbf{AB} = (4 – 1)\mathbf{i} + (0 – 2)\mathbf{j} + (-2 – 3)\mathbf{k} \]
\[ \mathbf{AB} = 3\mathbf{i} – 2\mathbf{j} – 5\mathbf{k} \]
No laila, ʻo ke vector e hoʻohui ana i ke kiko A iā B ʻo \( 3\mathbf{i} – 2\mathbf{j} – 5\mathbf{k} \).
Nīnau 2
Inā aia kahi kiko P ma \((2, 3)\) ma ka mokulele 2D, e ʻimi i ka lōʻihi (norm) o ka vector kūlana \(\mathbf{r_P}\).
Kūkākūkā:
ʻO ke vector kūlana o ke kiko P:
\[ \mathbf{r_P} = 2\mathbf{i} + 3\mathbf{j} \]
Hiki ke helu ʻia ka lōʻihi o ka vector kūlana \(\mathbf{r_P}\) me ka hoʻohana ʻana i ke ʻano vector norm (a i ʻole length):
\[ \| \mathbf{r_P} \| = \sqrt{x^2 + y^2} \]
E pani i nā waiwai o \(x\) a me \(y\):
\[ \| \mathbf{r_P} \| = \sqrt{2^2 + 3^2} \]
\[ \| \mathbf{r_P} \| = \sqrt{4 + 9} \]
\[ \| \mathbf{r_P} \| = \sqrt{13} \]
No laila, ʻo ka lōʻihi o ka vector kūlana \(\mathbf{r_P}\) ʻo \(\sqrt{13}\).
Nīnau 3
Manaʻo ʻia aia kahi kiko Q ma \( (5, -4, 2) \). E huli i ke kihi ma waena o ka vector kūlana \(\mathbf{r_Q}\) a me ke axis \(x\).
Kūkākūkā:
ʻO ke vector kūlana o ke kiko Q:
\[ \mathbf{r_Q} = 5\mathbf{i} – 4\mathbf{j} + 2\mathbf{k} \]
No ka loaʻa ʻana o ke kihi ma waena o ka vector \(\mathbf{r_Q}\) a me ke axis \(x\), hiki iā mākou ke hoʻohana i ke kumumanaʻo o ka huahana kiko. ʻO ka mea mua, hoʻoholo mākou i ka huahana kiko ma waena o \(\mathbf{r_Q}\) a me \(\mathbf{i}\):
\[ \mathbf{r_Q} \cdot \mathbf{i} = 5\mathbf{i} \cdot \mathbf{i} + (-4\mathbf{j} \cdot \mathbf{i}) + 2\mathbf{k} \cdot \mathbf{i} \]
ʻOiai ʻo \(\mathbf{i} \cdot \mathbf{i} = 1\), \(\mathbf{j} \cdot \mathbf{i} = 0\), a me \(\mathbf{k} \cdot \mathbf{i} = 0\), a laila:
\[ \mathbf{r_Q} \cdot \mathbf{i} = 5 \]
ʻO ke kūlana maʻamau o \(\mathbf{r_Q}\):
\[ \| \mathbf{r_Q} \| = \sqrt{5^2 + (-4)^2 + 2^2} \]
\[ \| \mathbf{r_Q} \| = \sqrt{25 + 16 + 4} \]
\[ \| \mathbf{r_Q} \| = \sqrt{45} \]
\[ \| \mathbf{r_Q} \| = 3\sqrt{5} \]
ʻO ke kūlana maʻamau o \(\mathbf{i}\) he 1, no ka mea, he vector unit ʻo \(\mathbf{i}\).
Ke hoʻohana nei i ke ʻano huahana kiko e ʻike ai i ke kihi \(\theta\):
\[ \mathbf{r_Q} \cdot \mathbf{i} = \| \mathbf{r_Q} \| \| \mathbf{i} \| \cos\theta\]
\[ 5 = 3\sqrt{5} \cos\theta \]
\[ \cos\theta = \frac{5}{3\sqrt{5}} \]
\[ \cos\theta = \frac{5}{3\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} \]
\[ \cos\theta = \frac{5\sqrt{5}}{15} \]
\[ \cos\theta = \frac{\sqrt{5}}{3} \]
No laila, ʻo ke kihi \(\theta\) ma waena o ka vector kūlana \(\mathbf{r_Q}\) a me ke axis \(x\):
\[ \theta = \cos^{-1} \left(\frac{\sqrt{5}}{3}\right) \]
Ka hopena
He kuleana koʻikoʻi ko nā vectors kūlana i ka ʻepekema a me ka ʻenekinia, ʻoiai hoʻi i ke kaha kiʻi ʻana i ke kūlana o nā mea i loko o ka lewa hoʻonohonoho. Hōʻike nā hiʻohiʻona ma luna pehea e helu ai i nā vectors kūlana, ko lākou lōʻihi, a me nā kihi ma waena o lākou a me nā axis hoʻonohonoho. He mea waiwai nui ka hoʻomaopopo ʻana i kēia mau manaʻo kumu i ka hoʻoponopono ʻana i nā pilikia like ʻole e pili ana i ka lewa a me nā hoʻonohonoho i ka makemakika a me ka physics.
