Ngā Wāhanga Wetereo: Ngā Kaupapa Taketake, Ngā Whakamāramatanga, me Ngā Whakamahinga
He ariā taketake ngā whārite i roto i te pāngarau, te ahupūngao, me te hangarau. I roto i ngā momo mara pūtaiao, ka whakamahia pinepine ēnei hei whakaahua i ngā rahinga he nui, he ahunga hoki. I roto i tēnei tuhinga, ka tūhuratia e tātou ngā wāhanga o te whārite: te tautuhi i te āhua o te whārite, me pēhea te wehewehe i te whārite ki roto i ōna wāhanga, me te tūhura i ngā tono me ngā pānga o ngā whārite i roto i te oranga o ia rā me te pūtaiao.
Te Mārama ki ngā Wētere
He rahinga te whārite, ehara i te mea he uara anake (rahi) engari he ahunga anō hoki. Kāore i rite ki ngā tauine, he uara anake (pērā i te pāmahana, i te papatipu rānei), e rua ngā āhuatanga matua o ngā whārite, ā, e whakamahia ana hei tohu i ngā āhuatanga e whai wāhi nui ana te ahunga, pērā i te tere, te kaha, me te nekehanga.
Mā te pāngarau, ka taea te whakaatu i tētahi whārite i roto i te wāhi rua-ahu (2D) pēnei \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \end{bmatrix}\), ko \(v_x\) me \(v_y\) ngā wāhanga o te whārite \(\mathbf{v}\) i roto i ngā ahunga-x me te ahunga-y. I roto i te wāhi toru-ahu (3D), ka taea te whakaatu i tētahi whārite pēnei \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}\).
Te Whakaaturanga Wetereo me ngā Wāhanga
Hei mārama ki te ariā o ngā wāhanga whārite, me mōhio tātou ka taea te wehewehe i ngā whārite ki ngā wāhanga e rite ana ki ia tuaka taunga. Hei tauira, i roto i te wāhi rua-ahu, ka taea te wehewehe i te whārite \(\mathbf{v}\) ki ngā wāhanga e rua: \(v_x\) (te wāhanga i te ahunga-x) me \(v_y\) (te wāhanga i te ahunga-y).
I te taha āhuahanga, ki te tuhia he ira i runga i te papa taunga Cartesian, ka taea te whakarite ki tētahi pere e tohu ana mai i te pūtake \((0,0)\) ki te pūwāhi \((v_x, v_y)\). Ka taea te kite i ngā wāhanga \(v_x\) me \(v_y\) hei roa o ngā toronga o te ira ki runga i ngā tuaka-x me te-y.
I roto i te wāhi ahu-toru, ka taea te wehewehe i tētahi whārite kia toru ngā wāhanga: \(v_x\) (te wāhanga ahunga-x), \(v_y\) (te wāhanga ahunga-y), me \(v_z\) (te wāhanga ahunga-z). Arā, ka taea te whakaatu i tētahi whārite i roto i te wāhi ahu-toru mā te pere e tohu ana mai i te pūtake \((0,0,0)\) ki te pūwāhi \((v_x, v_y, v_z)\).
Te Rahi me te Aronga o ngā Wētera
Hei tatau i te rahi, te roa rānei o tētahi whārite \(\mathbf{v}\), ka whakamahia e mātou te tātai:
\[
|\mathbf{v}| = \sqrt{v_x^2 + v_y^2}
\]
mō te wāhi ahu-rua, ā:
\[
|\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}
\]
mō te wāhi ahu-toru. E kiia ana tēnei rahinga whārite ko tōna rahi, ā, e tohu ana i te roa o te whārite.
Ka taea te whakaatu i te ahunga o tētahi whārite mā te whakamahi i tōna koki e pā ana ki ngā tuaka taunga. I roto i te wāhi rua-ahu, ka taea te tatau i te ahunga o tētahi whārite \(\mathbf{v}\) e hanga ana i tētahi koki \(\theta\) me te tuaka-x mā te whakamahi i te ine whārite:
\[
\theta = \tan^{-1}\left(\frac{v_y}{v_x}\right)
\]
I roto i te wāhi toru-ahu, he uaua ake te whakatau i te ahunga, nā te mea me whai whakaaro tātou ki ngā koki o ia tuaka taunga. I te nuinga o te wā, ka whakamahia he pūnaha porowhita hei whakaatu i te ahunga i roto i te wāhi toru-ahu.
Ngā Mahi i runga i ngā Wetere
Tāpiri me te Tango
Mā te tāpiri i ngā whārite e rua ka taea te mahi mā te tāpiri i ngā wāhanga takitahi o ngā whārite e rua. Hei tauira, mēnā ko \(\mathbf{u} = \begin{bmatrix} u_x \\ u_y \end{bmatrix}\) me \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \end{bmatrix}\), kātahi:
\[
\mathbf{u} + \mathbf{v} = \begin{bmatrix} u_x + v_x \\ u_y + v_y \end{bmatrix}
\]
Ka tatauhia te tangohanga whārite i tētahi ara ōrite:
\[
\mathbf{u} – \mathbf{v} = \begin{bmatrix} u_x – v_x \\ u_y – v_y \end{bmatrix}
\]
Whakareatanga Tauine
Ko te whakarea i tētahi irahiko ki tētahi tauine (he tau kotahi) ka mahia mā te whakarea i ia wāhanga o te irahiko ki te tauine. Hei tauira, mēnā he tauine te \(k\) ā, ko \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \end{bmatrix}\), kātahi:
\[
k \cdot \mathbf{v} = \begin{bmatrix} k \cdot v_x \\ k \cdot v_y \end{bmatrix}
\]
Te Whakareatanga Ira me te Whakawhiti
I roto i te wāhi ahu-toru, e rua ngā momo whakarea whārite: te whakarea ira me te whakarea whakawhiti.
1. Te Whakareatanga Ira:
Ko te hua ira o ngā whārite e rua \(\mathbf{u} = \begin{bmatrix} u_x \\ u_y \\ u_z \end{bmatrix}\) me \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}\) e whakamāramatia ana ko:
\[
\mathbf{u} \cdot \mathbf{v} = u_x v_x + u_y v_y + u_z v_z
\]
Ko te hua o te hua ira he tauine. He maha ngā wā ka whakamahia te hua ira hei whakatau i te nui o te whakarara, te tūhonohono rānei o ngā whārite e rua tetahi ki tetahi.
2. Whakareatanga whakawhiti:
Ko te hua whakawhiti o ngā whārite e rua i roto i te wāhi toru-ahu ka puta he whārite hou e poutū ana ki ngā whārite taketake e rua. Mena \(\mathbf{u} = \begin{bmatrix} u_x \\ u_y \\ u_z \end{bmatrix}\) me \(\mathbf{v} = \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}\), ka tautuhia te hua whakawhiti penei:
\[
\mathbf{u} \times \mathbf{v} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
u_x me u_y me u_z \\
v_x me v_y me v_z
\end{vmatrix}
\]
Whakaōritenga Wetereo
Ko te whakataurite ko te tukanga huri i tētahi whārite ki tētahi whārite kotahi (he whārite he roa 1) me te ahunga kotahi. Ka whiwhihia te whārite kotahi \(\mathbf{\hat{v}}\) o \(\mathbf{v}\) mā te wehewehe i ia wāhanga mā te roa (rahi) o te whārite:
\[
\mathbf{\hat{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}
\]
Ngā Whakamahinga o ngā Weka i roto i te Oranga o Ia Rā me te Pūtaiao
He maha ngā whakamahinga o ngā whārite i roto i te oranga o ia rā me te pūtaiao. Anei ētahi tauira:
1. Ahupūngao:
I roto i te ahupūngao, ka whakamahia ngā whārite hei whakaahua i ngā rahinga rerekē pērā i te tere, te whakaterenga, te kaha, me te nekehanga. Hei tauira, ka taea te tātari i te nekehanga o tētahi mea mā te whakamahi i ngā whārite tere me te whakaterenga.
2. Tikanga:
I roto i te hangarau, ka whakamahia ngā whārite mō te tātari hanganga, te hoahoa mīhini, me ētahi atu tono hangarau. Hei tauira, ko te tātari ahotea me te taumahatanga i roto i tētahi rauemi he maha ngā wā ka uru atu ki te whakamahinga o ngā whārite.
3. Whakairoiro Rorohiko:
Ka whakamahia hoki ngā wetere i roto i ngā whakairoiro rorohiko hei whakaahua i te tūranga, te aronga, me te nekehanga o ngā mea. I roto i te hōtaka whakairoiro, ka whakamahia ngā wetere mō ngā panonitanga pēnei i te whakawhiti, te hurihanga, me te tauine.
4. Whakatere:
Ka whakamahia ngā whārite i roto i ngā pūnaha whakatere hei whakatau i te ahunga me te tawhiti i waenga i ngā pūwāhi e rua. Ka whakamahia e te GPS me ētahi atu pūnaha whakatere ngā whārite hei tatau i ngā huarahi me te tohutohu i ngā kaiwhakamahi.
5. Ōhanga:
I roto i te ōhanga, ka taea te whakamahi i ngā whārite hei whakaahua i ngā manakohanga a te kaihoko, i ngā putea haumi rānei. He maha hoki ngā wā ka uru atu ki te tātari raraunga maha-taurangi te whakamahinga o ngā whārite.
Whakamutunga
He ariā tino nui, he ariā whānui hoki ngā whārite i roto i te pāngarau me te maha atu o ngā mara pūtaiao. Mā te mārama ki ngā wāhanga o ngā whārite me ngā mahi ka taea te mahi ki runga i a rātou, ka taea e tātou te whakamahi i tēnei ariā hei whakaoti i te whānuitanga o ngā raruraru mahi me ngā raruraru pūtaiao. Mā te kaha o tā rātou whakaaturanga pāngarau, ka whakaratohia e ngā whārite he taputapu whai hua mō te whakaahua me te tātari i te whānuitanga o ngā āhuatanga e pā ana ki ngā rahinga me ngā ahunga.
