Tauira Pātai e Matapaki ana i te Tāpiritanga o ngā Wēka e Rua mā te Whakamahi i te Tikanga Whakarara
He ariā nui te tāpiritanga whārite i roto i te ahupūngao me te pāngarau, e whakamahia ana hei whakaahua i ngā āhuatanga taiao me ngā raruraru o ia rā. He maha ngā tikanga hei tāpiri i ngā whārite e rua, ko tētahi o ēnei ko te tikanga whakarara. Ehara i te mea he māmā noa iho tēnei tikanga engari he whakaaturanga kaha hoki mō te hononga o ngā whārite e rua hei hanga i tētahi whārite hua. I roto i tēnei tuhinga, ka tirohia e tātou ētahi tauira o te tāpiritanga whārite mā te whakamahi i te tikanga whakarara, me ā rātou otinga.
He aha te Wetere?
I mua i te urunga atu ki ngā tauira raruraru, me mārama tātou ki te whakamāramatanga taketake o te whārite. Ko te whārite he rahinga kei a ia te rahi (te roa) me te ahunga. Ko ngā tauira matarohia o ngā whārite ko te tere, te whakaterenga, te kaha, me te nekehanga. Ka taea te whakaatu i te whārite hei ōna wāhanga (i, j, k) i roto i ngā taunga Cartesian, hei tōna roa me te ahunga (te koki) rānei.
Tikanga Whakarara
Ko te tikanga whakarara tētahi huarahi hei tāpiri i ngā whārite e rua. I tēnei tikanga, ka whakaatuhia e mātou ngā whārite e rua hei taha e rua o te whakarara. Ko te whārite ka puta ko te hauroki o te whakarara e tīmata ana mai i te pūwāhi tīmatanga o ngā whārite e rua. Mā te pāngarau, mēnā e rua ā tātou whārite \(\vec{A}\) me \(\vec{B}\), ko te hua ko \( \vec{R} = \vec{A} + \vec{B} \).
Ko te tikanga taahiraa-i-te-taahiraa mō te whakamahi i te tikanga whakarara-whakarara koia tēnei:
1. Tuhia te whārite \(\vec{A}\) mai i te pūwāhi tīmatanga.
2. Mai i te pito o te whārite \(\vec{A}\), tuhia te whārite \(\vec{B}\).
3. Tuhia he rārangi whakarara ki te whārite \(\vec{B}\) mai i te pūwāhi tīmatanga \(\vec{A}\).
4. Tuhia he rārangi whakarara ki te whārite \(\vec{A}\) mai i te pito o te whārite \(\vec{B}\).
5. Tuhia he whakarara mai i te pūwāhi tīmatanga ki te kokonga whakarara hei tiki i te whārite hua \(\vec{R}\).
Ngā Pātai Tauira me te Kōrero
Pātai 1
Me kī he rua ā tātou pūwero \(\vec{A}\) me \(\vec{B}\):
– Ko te roa (rahi) o \(\vec{A}\) he 5 waeine, ā, ko te ahunga he 0° (i te taha rānei o te tuaka-x pai),
– E 3 ngā waeine o te roa o \(\vec{B}\), ā, e 90° te ahunga (i te taha rānei o te tuaka-y pai).
He aha te uara hua o te tāpiri i ēnei whārite e rua mā te whakamahi i te tikanga whakarara?
Kōrero:
1. Tuhia te whārite \(\vec{A}\) i te tuaka-x pai me te roa o te 5 waeine.
2. Mai i te pito o te whārite \(\vec{A}\), tuhia te whārite \(\vec{B}\) i te tuaka-y pai me te roa o te 3 waeine.
3. Mai i te pūwāhi tīmatanga \(\vec{A}\), tuhia he rārangi whakarara ki \(\vec{B}\).
4. Mai i te pito o \(\vec{B}\), tuhia he rārangi whakarara ki \(\vec{A}\).
5. Ko te hua he whakarara me te whakarara koia te hua o te whārite \(\vec{R}\).
Nā te mea he poutū te tūnga o \(\vec{A}\) me \(\vec{B}\) tetahi ki tetahi, ka taea e tātou te whakamahi i te ariā Pythagorean hei tatau i te roa o te whārite hua:
\[ R = \sqrt{A^2 + B^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \tata ki te 5.83 \]
Ka taea te tatau i te ahunga o te ira hua mā te whakamahi i te ine whārite. Mena ko \(\theta\) te koki i waenganui i te hua me \(\vec{A}\):
\[ \tan(\theta) = \frac{B}{A} = \frac{3}{5} \]
nā reira:
\[ \theta = \tan^{-1}\left(\frac{3}{5}\right) \approx 30.96^\circ \]
Nō reira, ko te rahi o te whārite hua \(\vec{R}\) he tata ki te 5.83 waeine, ā, ko te ahunga mai i \(\vec{A}\ he tata ki te 30.96°.
Pātai 2
E rua ngā pūwero \(\vec{C}\) me \(\vec{D}\) e hoatu ana penei:
– \(\vec{C}\) me te roa o te 4 waeine me te ahunga o te 45°.
– \(\vec{D}\) me te roa o ngā waeine e 6, me te ahunga o te 120°.
Tāutuhia te hua o te whārite \(\vec{R}\) mai i te tāpiritanga o ngā whārite e rua.
Kōrero:
Hei tāpiri i ngā wekere e rua kāore e poutū tetahi ki tetahi, kāore hoki i te rerekē ngā āhua, ka taea e koe te whakamahi i ngā wāhanga Cartesian.
1. Wāwāhia a \(\vec{C}\) me \(\vec{D}\) kia rite ki ngā wāhanga x me y.
Mō \(\vec{C}\):
\[ C_x = C \cos(45^\circ) = 4 \cos(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2} \approx 2.83 \]
\[ C_y = C \sin(45^\circ) = 4 \sin(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2} \approx 2.83 \]
Mō \(\vec{D}\):
\[ D_x = D \cos(120^\circ) = 6 \cos(120^\circ) = 6 \cdot (-\frac{1}{2}) = -3 \]
\[ D_y = D \sin(120^\circ) = 6 \sin(120^\circ) = 6 \cdot \frac{\sqrt{3}}{2} = 3\sqrt{3} \approx 5.20 \]
2. Tāpirihia ngā wāhanga x me y o ngā whārite e rua:
\[ R_x = C_x + D_x = 2.83 + (-3) = -0.17 \]
\[ R_y = C_y + D_y = 2.83 + 5.20 = 8.03 \]
3. Tātaihia te rahi me te ahunga o te whārite hua \(\vec{R}\):
R = \sqrt{R_x^2 + R_y^2} = \sqrt{(-0.17)^2 + 8.03^2} = \sqrt{0.03 + 64.48} = \sqrt{64.51} \approx 8.03 \]
\[ \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{8.03}{-0.17}\right) \approx \tan^{-1}(-47.24) \]
Nā te mea he kino te hua, ka tāpirihia e tātou te 180° kia whiwhi ai i te koki i roto i te pūnaha hauwhā tika:
\[ \theta \approx \tan^{-1}(47.24) + 180^\circ \approx 271.93^\circ \]
Nō reira, ko te rahi o te whārite hua \(\vec{R}\) he tata ki te 8.03 waeine, ā, ko te ahunga he tata ki te 271.93°, ka taea rānei te kī he tata ki te 91.93° mai i te tuaka-x kino i te hauwhā tuawhā.
Te Katinga
He huarahi whai hua, ā, he huarahi ātaahua hoki te tikanga whakarara ki te tāpiri i ngā whārite e rua. Ahakoa he māmā noa iho tēnei tikanga mō ngā whārite māmā, he mea nui kia mārama, mō ngā whārite uaua ake, me whakamahi pinepine tātou i ngā wāhanga Cartesian me ngā tikanga taurangi matatau ake hei whiwhi hua tika. Ko te tumanako, mā ngā tauira i runga ake nei ka mārama te whakamahinga o tēnei tikanga i roto i ngā āhuatanga rerekē.
