Tātaritanga Wetereo i te Wāhi
Ko te tātaritanga whārite i te wāhi he peka o te pāngarau e aro ana ki te ako i ngā whārite me ā rātou mahi i te wāhi toru-ahu (3D). Ko te whārite he rahinga he nui, he ahunga hoki, he rerekē ki te tauine, he nui anake tōna. Ka whakamahia ngā whārite i te wāhi i roto i te whānuitanga o ngā kaupapa ako, mai i te ahupūngao ki te pūtaiao rorohiko, ā, he taputapu nui i roto i te tātaritanga ā-ira, te kinematics, me te dynamics.
Ariā Taketake o ngā Wētera
Ka taea te whakaatu i tētahi whārite i roto i te wāhi toru-ahu hei v = (v₁, v₂, v₃), ko v₁, v₂, me v₃ ngā wāhanga o te whārite i ngā ahunga x, y, me z. Ko te whakaaturanga whakairoiro o tētahi whārite he pere i tangohia mai i te pūtake (0, 0, 0) ki te pūwāhi (v₁, v₂, v₃). Ka taea te tatau i te roa o tētahi whārite (rahi) mā te whakamahi i te tātai:
\[ \| \mathbf{v} \| = \sqrt{v_1^2 + v_2^2 + v_3^2} \]
Ngā Mahi Taketake i runga i ngā Wetere
1. Tāpiri me te Tango
E rua ngā whārite e taea te tāpiri, te tango rānei mā te tāpiri, te tango rānei i ō rāua wāhanga:
\[ \mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, u_3 + v_3) \]
\[ \mathbf{u} – \mathbf{v} = (u_1 – v_1, u_2 – v_2, u_3 – v_3) \]
2. Te Whakarea mā te Tauine
Mena he tauine (tau tūturu) a c, ko te whakarea o te whārite v ki te tauine c koia tēnei:
\[ c\mathbf{v} = (cv_1, cv_2, cv_3) \]
3. Hua Ira
Ko te hua ira i waenganui i ngā whārite e rua u me v he tauine e tautuhia ana ko:
\[ \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \]
E whakaatu ana hoki tēnei hua ira mēnā he whakarara ngā whārite e rua, nā te mea he ōrite te hua ira o ngā whārite poutū (tūroa) e rua ki te kore.
4. Hua Whakawhiti
Ko te hua whakawhiti o ngā whārite e rua, arā, a u me a v, ka puta he whārite hou e hangai ana ki ngā whārite e rua. Ka whakaaturia pēnei:
\[ \mathbf{u} \times \mathbf{v} = \left( u_2v_3 – u_3v_2, u_3v_1 – u_1v_3, u_1v_2 – u_2v_1 \right) \]
Ngā Taupānga Tātari Wetereo
1. Kinematika
I roto i te kinematics, ka whakaahuahia te nekehanga o tētahi mea mā te whakamahi i ngā whārite tūnga, tere, me te whakaterenga. Hei tauira, ki te neke tētahi mea i roto i te wāhi 3D, ka taea te whakaahua i tōna tūranga i te wā t mā te whārite tūnga r(t). Ko te tere o te mea ko te pānga o te whārite tūnga e pā ana ki te wā:
\[ \mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt} \]
Ahakoa ko te whakaterenga te pānga o te wete tere:
\[ \mathbf{a}(t) = \frac{d\mathbf{v}(t)}{dt} \]
2. Ngā Āhuatanga
I roto i ngā mahi auaha, ka whakamahia te tātaritanga whārite hei tatau i ngā kaha e pā ana ki tētahi mea. Hei tauira, ka taea te whakaatu i te ture tuarua a Newton i roto i te āhua whārite penei:
\[ \mathbf{F} = m\mathbf{a} \]
ko F te kaha kupenga e pā ana ki te mea me te papatipu m, ā, ko a te whakaterenga o te mea.
3. Te aukumetanga hiko
He nui hoki te whakamahinga o te tātaritanga whārite i roto i te aukumetanga hiko. Hei tauira, ko te papa hiko E me te papa aukume B he whārite e whakawhirinaki ana ki tō rāua tūranga i te wāhi. Ko ngā whārite a Maxwell, e whakaahua ana i te whanaketanga o ngā papa hiko me ngā papa aukume, he whārite rerekētanga i roto i te āhua whārite.
4. Whakairoiro Rorohiko
I roto i ngā whakairoiro rorohiko me te pakiwaituhi, ka whakamahia ngā wetereo hei tohu i te tūranga, te whakatakotoranga, me te tauine o ngā mea i roto i te wāhi toru-ahu. Ka whakamahia ngā panonitanga āhuahanga pērā i te nekehanga, te hurihanga, me te tauine ki ēnei mea mā te whakamahi i ngā matihiko panoni e mahi ana i runga i ngā wetereo tūranga o ngā pūwāhi o te mea.
Huringa Raina
Ko te panonitanga rārangi he mahi e hono ana i tētahi whārite ki tētahi atu whārite i te wāhi kotahi, i roto i te āhua rārangi. Ka taea te whakaatu i tēnei panonitanga mā te whakamahi i tētahi matihiko. Me kī ko T he panonitanga rārangi, ā, ko A tōna matihiko. Mena he whārite a v, ka taea te tuhi i te panonitanga rārangi penei:
\[ T(\mathbf{v}) = \mathbf{A} \mathbf{v} \]
Ko ngā panonitanga rārangi ko te hurihanga, te whakaata, te whakawhānui, me te kutikuti.
Matrix Whakawhiti
Ka taea te whakaatu i ia panonitanga rārangi mā te whakamahi i tētahi matihiko. Anei ētahi tauira o ngā matihiko panonitanga:
1. Hurihanga
Ko te hurihanga huri noa i te tuaka-z mā te koki θ e whakaatuhia ana e te matihiko:
\[
\mathbf{R}_z(\theta) = \begin{pmatrix}
\cos \theta & -\sin \theta & 0 \\
\sin \theta & \cos \theta & 0 \\
0 & 0 & 1
\end{pmatrix}
\]
2. Whakaaroaro
Ko te whakaata i te papa xy e whakaatuhia ana e te matihiko:
\[
\mathbf{R}_{xy} = \begin{pmatrix}
1 me te 0 me te 0
0 me te 1 me te 0
0 me te 0 me te -1
\end{pmatrix}
\]
3. Tauine
Ko te panonitanga tauine me te tauwehe s i ngā ahunga katoa (isotropic) e whakaatuhia ana e te matihiko:
\[
\mathbf{S}(s) = \begin{pmatrix}
s me te 0 me te 0 \\
0 me ngā me te 0 \\
0 me te 0 me ngā
\end{pmatrix}
\]
Ngā Huarahi Āhua me ngā Uara Āhua
I roto i te horopaki o ngā panonitanga rārangi, he ariā nui ngā eigenvectors me ngā eigenvalues. Mēnā he matihiko panonitanga rārangi a A, he eigenvalue a λ, ā, he eigenvector a v, kātahi:
\[ \mathbf{A} \mathbf{v} = \lambda \mathbf{v} \]
Ko te eigenvector he eigen e tiakina ana tōna ahunga me tōna tauine i muri i te panonitanga, ko te eigenvalue ia he tauwehe o taua tauine. Mā te tātari i ngā eigenvector me ngā eigenvalue ka taea e tātou te mārama ki ngā āhuatanga o ngā matrices me ngā panonitanga raina matatini.
Whakamutunga
He taputapu kaha, he taputapu maha hoki te tātaritanga whārite i roto i te pāngarau me te pūtaiao. Mā te mārama ki ngā mahi whārite taketake me ā rātou tono, ka taea e tātou te whakaoti i ngā momo raruraru maha i roto i te ahupūngao, te hangarau, te whakairoiro rorohiko, me te maha atu o ngā mara. Mā te mōhio ki ngā ariā o ngā panonitanga rārangi, ngā hua ira, ngā hua whakawhiti, me ngā whārite eigen me ngā uara eigen ka taea e tātou te tātari me te whakatauira i ngā pūnaha tino uaua me te whai hua me te whānui.
