Tātai Wētera Hua: Ariā, Tikanga, me ngā Tauira Raruraru
Ko te whārite he rahinga kei a ia te rahi me te ahunga. I roto i te ahupūngao me te pāngarau, he maha ngā whakamahinga o ngā whārite hei whakaahua i ngā āhuatanga rerekē pēnei i te tere, te kaha, me te nekehanga. Ko te tatau i te whārite hua, te tapeke o ngā whārite e rua, neke atu rānei, he pūkenga nui e whakamahia ana i roto i ngā tono pūtaiao me ngā tono hangarau. Ka matapakihia e tēnei tuhinga te ariā taketake o ngā whārite, ngā tikanga mō te tatau i te whārite hua, me te whakarato i ētahi tauira raruraru hei whakamārama i te māramatanga.
Te Mārama ki ngā Wētera me ngā Wētera Hua
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He mea pāngarau te whārite, e rua ōna āhuatanga matua:
1. Te Rahi: Te rahi o te uara whārite.
2. Aronga: Ko te ahunga o te whārite e tohu ana i te whakatakotoranga o te whārite i te wāhi.
He maha ngā wā ka whakaaturia ngā whārite hei pere, ko te roa o te pere te tohu o te rahi, ā, ko te ahunga o te pere te tohu o te ahunga o te whārite.
Te Hua o te Wētera
Ko te whārite hua he whārite kotahi e tohu ana i te huinga o ngā whārite e rua, neke atu rānei. Ko te tukanga tāpiri whārite e mōhiotia ana ko te "tāpiri whārite". He maha ngā tikanga ka taea te whakamahi hei tatau i te whārite hua, tae atu ki ngā tikanga whakairoiro me ngā tikanga tātari.
Tikanga Tātai Hua Wekita
Tikanga Whakairoiro
Ko te tikanga whakairoiro ko te whakaatu i ngā whārite i runga i te āhua ā-ira, me te whakamahi i ngā ture tāpiri whārite hei kimi i te hua. Ko ngā ture matua e rua o te tikanga whakairoiro ko:
1. Tikanga Tapatoru: I tēnei tikanga, ka tangohia te whārite tuarua mai i te pito o te whārite tuatahi. Ko te whārite hua ko te whārite i tangohia mai i te pūwāhi tīmatanga o te whārite tuatahi ki te pito o te whārite tuarua.
2. Tikanga Porowhita: Ka whakamahia tēnei tikanga hei tāpiri i ngā whārite neke atu i te rua. Ka tuhia ngā whārite i runga i te raupapa mai i te pito ki te pito, ā, ko te whārite hua ko te whārite e hono ana i te pūwāhi tīmatanga o te whārite tuatahi ki te pito o te whārite whakamutunga.
Tikanga Tātari
Ko te tikanga tātari e whakamahi ana i te pāngarau me te ine whārite hei tatau i te hua o te wetereo. Ko ngā tikanga matua e rua o te tikanga tātari ko:
1. Tikanga Wāhanga: I tēnei tikanga, ka wehea ia whārite ki ōna wāhanga i te taha o ngā tuaka-x me te tuaka-y. Kātahi ka tāpirihia ēnei wāhanga kia whiwhi ai i ngā wāhanga o te whārite hua. Hei whakamutunga, ka tatauhia te whārite hua mā te whakamahi i te ariā Pythagorean me te trigonometry.
2. Tikanga Kōsine: Ka whakamahia tēnei tikanga ina mōhiotia te rahi o ngā whārite e rua me te koki i waenganui i a rāua. Ka whakamahia te tātai kōsine hei tatau i te rahi o te whārite hua.
Ngā Tātai Wētera Hua
Tikanga Wāhanga
Mō ngā whārite e rua \(\mathbf{A}\) me \(\mathbf{B}\) me ngā wāhanga:
\[
\mathbf{A} = A_x \pōtae{i} + A_y \pōtae{j}
\]
\[
\mathbf{B} = B_x \pōtae{i} + B_y \pōtae{j}
\]
Ko te hua o te whārite \(\mathbf{R}\) ko:
\[
\mathbf{R} = \mathbf{A} + \mathbf{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \pōtae{j}
\]
Ka taea te tatau i te rahi o te whārite hua \(\mathbf{R}\) mā te whakamahi i te ariā Pythagorean:
\[
|\mathbf{R}| = \sqrt{(A_x + B_x)^2 + (A_y + B_y)^2}
\]
Ko te ahunga o te whārite hua ka whakatauhia e te koki \(\theta\) i hangaia me te tuaka-x:
\[
\theta = \tan^{-1}\left(\frac{A_y + B_y}{A_x + B_x}\right)
\]
Tikanga Kōsina
Mena he rahi o ngā whārite e rua \(\mathbf{A}\) me \(\mathbf{B}\) \(A\) me \(B\) ā, he koki \(\theta\) kei waenganui i a rāua, ko te rahi o te whārite hua \(\mathbf{R}\) ko:
\[
|\mathbf{R}| = \sqrt{A^2 + B^2 + 2AB \cos \theta}
\]
Ka taea te tatau i te ahunga o te whārite hua mā te whakamahi i te tātai pākoki:
\[
\tan \alpha = \frac{B \sin \theta}{A + B \cos \theta}
\]
Ko \(\alpha\) te koki i hangaia e te ira hua me te ira \(\mathbf{A}\).
Tauira o te Raru Whakapūtanga Hua
Tauira Pātai 1: Tikanga Wāhanga
Pātai:
E rua ngā whārite \(\mathbf{A}\) me \(\mathbf{B}\) kei a rātou ngā wāhanga e whai ake nei:
\[
\mathbf{A} = 3\pōtae{i} + 4\pōtae{j}
\]
\[
\mathbf{B} = 1\pōtae{i} + 2\pōtae{j}
\]
Tātaihia te hua o te whārite \(\mathbf{R}\).
Otinga:
1. Tāpirihia ngā wāhanga ki ngā tuaka x me y:
\[
R_x = A_x + B_x = 3 + 1 = 4
\]
\[
R_y = A_y + B_y = 4 + 2 = 6
\]
2. Tātaihia te rahi o te whārite hua:
\[
|\mathbf{R}| = \sqrt{R_x^2 + R_y^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 7,21
\]
3. Tātaihia te ahunga o te whārite hua:
\[
\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{6}{4}\right) = \tan^{-1}(1,5) = 56,31^\circ
\]
Nō reira, ko te rahi o te whārite hua \(\mathbf{R}\) he 7,21, ā, ko te ahunga he 56,31 nekehanga ki te tuaka-x.
Tauira Pātai 2: Tikanga Kōsina
Pātai:
E rua ngā whārite \(\mathbf{A}\) me \(\mathbf{B}\) he rahi \(A = 5\) ngā waeine, \(B = 7\) ngā waeine, ā, ko te koki i waenganui i a rāua he 60°. Tātaihia te rahi o te whārite hua \(\mathbf{R}\).
Otinga:
1. Whakamahia te tātai kosinī hei tatau i te rahi o te whārite hua:
\[
|\mathbf{R}| = \sqrt{A^2 + B^2 + 2AB \cos \theta}
\]
\[
|\mathbf{R}| = \sqrt{5^2 + 7^2 + 2 \cdot 5 \cdot 7 \cdot \cos 60^\circ}
\]
\[
|\mathbf{R}| = \sqrt{25 + 49 + 70 \cdot 0,5}
\]
\[
|\mathbf{R}| = \sqrt{25 + 49 + 35}
\]
\[
|\mathbf{R}| = \sqrt{109} = 10,44 \, \text{unit}
\]
Nō reira, ko te rahi o te whārite hua \(\mathbf{R}\) he 10,44 waeine.
Tauira 3: Hua o ngā Wētera e Toru
Pātai:
E toru ngā whārite \(\mathbf{A}\), \(\mathbf{B}\), me \(\mathbf{C}\) kei a rātou ngā wāhanga e whai ake nei:
\[
\mathbf{A} = 2\pōtae{i} + 3\pōtae{j}
\]
\[
\mathbf{B} = -1\pōtae{i} + 4\pōtae{j}
\]
\[
\mathbf{C} = 3\pōtae{i} – 2\pōtae{j}
\]
Tātaihia te hua o te whārite \(\mathbf{R}\).
Otinga:
1. Tāpirihia ngā wāhanga ki ngā tuaka x me y:
\[
R_x = A_x + B_x + C_x = 2 – 1 + 3 = 4
\]
\[
R_y = A_y + B_y + C_y = 3 + 4 – 2 = 5
\]
2. Tātaihia te rahi o te whārite hua:
\[
|\mathbf{R}| = \sqrt{R_x^2 + R_y^2} = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} = 6,4
\]
3. Tātaihia te ahunga o te whārite hua:
\[
\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{5}{4}\right) = \tan^{-1}(1,25) = 51,34^\circ
\]
Nō reira, ko te hua o te whārite \(\mathbf{
Ko te rahi o R}\) he 6,4, ā, ko te ahunga ki te tuaka-x he 51,34 nekehanga.
Whakamutunga
He pūkenga nui te tatau i te hua o tētahi whārite i roto i te ahupūngao me te pāngarau. Mā te whakamahi i ngā tikanga whakairoiro, tātari rānei, ka taea e tātou te whakatau i te hua o ngā whārite e rua, neke atu rānei. Ko te tikanga o ngā wāhanga me te tikanga o ngā cosine he tikanga matua e rua i roto i ngā tatau tātari e āhei ai tātou ki te tatau tika i te rahi me te ahunga o te whārite hua. E whakaatu ana ngā tauira i runga ake nei i te whakamahinga mahi o ēnei ariā, e āwhina ana i a tātou ki te mārama me te whakamahi i ngā whārite i roto i ngā āhuatanga pūtaiao me ngā āhuatanga hangarau.
