Ngā Kupu me ngā Momo o te Tuhituhinga Wetereo
Ina kōrerohia te pāngarau, te ahupūngao, me te pūtaiao rorohiko, he mea nui te ariā o ngā whārite hei mārama. Ehara i te mea he ariā noa iho ngā whārite; he mea nui hoki i roto i ngā āhuatanga mahi maha, pērā i te tātari raraunga, te whakairoiro rorohiko, me ngā whakatauira ahupūngao. I roto i tēnei tuhinga, ka matapakihia e mātou ngā kupu me ngā tuhi whārite, kātahi ka tūhuratia ngā momo whārite e kitea ana i roto i ēnei marautanga.
Ngā Kupu me ngā Tuhituhinga o te Wetereo
1. Ngā Wēka me ngā Tauine
He mea pāngarau te whārite, he rahi, he ahunga hoki tōna. He rerekē, he uara kotahi te tauine, he rahi anake tōna, kāore he ahunga. Hei tauira, he tauine te tere o te 5 m/s me te kore e tohu i te ahunga, ā, he tauine te tere o te 5 m/s ki te rawhiti.
2. Te Tuhituhinga Wetere
Ko ngā whākarite he tohu i te nuinga o te wā mā te reta iti matotoru pēnei i te v , mā te pere rānei i runga ake i te reta pēnei i te \(\vec{v}\). Hei tauira, mēnā he whākarite v tā tātou ko ōna huānga ko \(v_1, v_2, v_3\), ka taea tēnei te tuhi penei:
\[ \vec{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} \]
Ko tētahi atu huarahi ki te tuhi i ngā whārite, inā koa i roto i ngā horopaki rua-ahu, toru-ahu rānei, ko te whakamahi i tētahi pūtake paerewa. Hei tauira:
\[ \vec{v} = v_1\hat{i} + v_2\hat{j} + v_3\hat{k} \]
ko \(\hat{i}, \hat{j}\), me \(\hat{k}\) ngā whārite wae i runga i ngā tuaka x, y, me z.
Ngā Momo o ngā Wetere
1. Te Wāhi Tūnga
Ko te ira tūnga he ira e whakaahua ana i te tūnga o tētahi pūwāhi i te wāhi e pā ana ki tētahi pūwāhi tohutoro, ko te nuinga ko te pūwāhi O (te pūtake). Mena he taunga (x, y, z) tō te pūwāhi P i te wāhi 3D, ka taea te whakaatu i te ira tūnga \(\vec{r}\) penei:
\[ \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \]
2. Te Wētera Nekehanga
E whakaahua ana te whārite nekehanga i te huringa o te tūranga o tētahi pūwāhi mai i tētahi tūranga ki tētahi atu. Me kī he taunga kei te pūwāhi A (x1, y1, z1) ā, he taunga kei te pūwāhi B (x2, y2, z2). Ka taea te tuhi i te whārite nekehanga \(\vec{d}\) mai i A ki B penei:
\[ \vec{d} = (x2 – x1)\pōtae{i} + (y2 – y1)\pōtae{j} + (z2 – z1)\pōtae{k} \]
3. Te Tere o te Wētere
Ko te tere he whārite e tohu ana i te tere o te huringa o te tūranga o tētahi mea mō ia wā. Mena he pānga o te tūranga e pā ana ki te wā te \(\vec{r}(t)\) , ko te whārite tere \(\vec{v}(t)\) te pānga o \(\vec{r}(t)\) e pā ana ki te wā t:
\[ \vec{v}(t) = \frac{d\vec{r}(t)}{dt} \]
4. Te Wētere Whakaterenga
Ko te whārite whakaterenga te pānga o te whārite tere e pā ana ki te wā. E tohu ana i te tere o te huringa o te tere o tētahi mea mō ia wae wā. Mena he mahi a \(\vec{v}(t)\) o te tere e pā ana ki te wā, ko te whārite whakaterenga \(\vec{a}(t)\) te pānga o \(\vec{v}(t)\):
\[ \vec{a}(t) = \frac{d\vec{v}(t)}{dt} \]
5. Te Whakakaha i te Waehere
E ai ki te ture tuarua a Newton, ko te kaha te hua o te papatipu me te whakaterenga. He whārite anō hoki te kaha nā te mea he nui, he ahunga hoki tōna. Mena ko m te papatipu, ā, ko \(\vec{a}\) te whārite whakaterenga, ka taea te whakaatu i te whārite kaha \(\vec{F}\) penei:
\[ \vec{F} = m\vec{a} \]
6. Te Wāhanga Wāhanga
Ko te ira kotahi he ira kotahi te rahi (roa). Ka taea te whiwhi i te ira kotahi o tētahi ira \(\vec{v}\) mā te wehewehe i te \(\vec{v}\) ki tōna rahi. Mena he \(||\vec{v}||\ te rahi o \(\vec{v}\), ka taea te tuhi i tōna ira kotahi penei:
\[ \hat{v} = \frac{\vec{v}}{||\vec{v}||} \]
7. Kore Wekete
Ko te kore-ira he ira e kore katoa ana ngā wāhanga, ā, ko te tikanga he \(\vec{0}\). Kāore he ahunga o tēnei ira, ā, ko tōna rahi he kore. Ko tētahi tauira i roto i te wāhi toru-ahu ko:
\[ \vec{0} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \]
8. Ngā Wētera Ā-Whārite
E kiia ana he orthogonal ngā whārite e rua mēnā he kore te hua o roto. Mēnā he whārite e rua a \(\vec{u}\) me \(\vec{v}\), ka orthogonal rāua mēnā:
\[ \vec{u} \cdot \vec{v} = 0 \]
9. Ngā Wētera Honohono me ngā Wētera Honohono
E kiia ana he hononga-taha ngā whārite e rua mēnā kei te rārangi tika kotahi, kei te rārangi whakarara rānei. Ka taea te whakaatu hei taurangi tauine o tētahi ki tētahi. Hei tauira:
\[ \vec{v} = k\vec{u} \]
mō ētahi tauine \(k\).
I taua wā anō, e kiia ana he papatahi ngā whārite e toru mēnā kei te papatahi kotahi. Ka taea te whakaatu hei huinga rārangi o ngā whārite e rua e toe ana.
Ngā Mahi i runga i ngā Wetere
1. Te Tāpiri me te Tangohanga o te Wētera
Ka mahia te tāpiritanga wetereo mā te tāpiri i ō rātou wāhanga e rite ana. Mena \(\vec{u} = \begin{pmatrix} u_1 \\ u_2 \\ u_3 \end{pmatrix}\) me \(\vec{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}\), kātahi:
\[ \vec{u} + \vec{v} = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \\ u_3 + v_3 \end{pmatrix} \]
Ka mahia te tangohanga mā te tango i ngā wāhanga e rite ana:
\[ \vec{u} – \vec{v} = \begin{pmatrix} u_1 – v_1 \\ u_2 – v_2 \\ u_3 – v_3 \end{pmatrix} \]
2. Whakarea Tauine
He mahi te whakarea tauine e whakamahi ana i tētahi whārite me tētahi tauine tauine (uara tau). Mena he tauine tauine a k, ā, \(\vec{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}\), kāti:
\[ k\vec{v} = \begin{pmatrix} kv_1 \\ kv_2 \\ kv_3 \end{pmatrix} \]
3. Hua ā-Roto (Hua Ira)
He tauine te hua o roto o ngā whārite e rua \(\vec{u}\) me \(\vec{v}\). Ka taea te tatau mā te:
\[ \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3 \]
4. Hua Whakawhiti
Ko te hua whakawhiti o ngā whārite e rua \(\vec{u}\) me \(\vec{v}\) ka puta he whārite hou e hangai ana ki ēnei whārite e rua. I roto i te wāhi toru-ahu, ka tatauhia tēnei penei:
\[ \vec{u} \times \vec{v} = \begin{vmatrix}
\hat{i} me \hat{j} me \hat{k} \\
u_1 me u_2 me u_3 \\
v_1 me v_2 me v_3
\end{vmatrix} \]
Whakamutunga
He mea nui te mārama ki ngā kupu me ngā tuhi o ngā vector, me ō rātou momo, i roto i ngā momo marautanga pūtaiao. Ehara i te mea he mea pāngarau noa iho ngā vector, engari he taputapu kaha anō hoki i roto i te ahupūngao, te hangarau, me te tātari hangarau mōhiohio. Mā te mārama pai ki ēnei ariā taketake, ka taea e tātou te whakatau ngāwari i ngā raruraru uaua i roto i te whānuitanga o ngā mara.
