Ngā Tauira Pātai e Matapaki ana i te Tāpiritanga Wekita mā te Wāhanga
He tikanga taketake te tāpiritanga whārite i roto i te ahupūngao me te pāngarau e whakamahia ana hei kimi i te hua o ngā whārite e rua, neke atu rānei. He tikanga tino whai hua te huarahi wāhanga-ā-wāhanga ki te whakaoti rapanga tāpiritanga whārite, inā koa ka mahi ki ngā whārite i roto i ngā āhuahanga e rua, e toru rānei. Ka whakamāramahia e tēnei tuhinga te ariā o te tāpiritanga whārite wāhanga-ā-wāhanga, me te whakarato i ētahi tauira rapanga me ngā otinga.
Te Ariā o te Tāpiritanga Wāhanga Wāhanga
Ka taea te wehewehe i ngā whārite katoa i roto i te wāhi rua-ahu (2D) kia rua ngā wāhanga: he wāhanga x (whakapae) me te wāhanga y (poutū). I roto i ngā āhuatanga toru (3D), he wāhanga anō kei ngā whārite, arā, te wāhanga z (hohonu).
Mehemea e rua ā tātou whārite A me B. Ka taea te whakaatu i ngā wāhanga o ēnei whārite penei:
– Kei te Vector A ngā wāhanga \(A_x\) me \(A_y\) i te 2D (me \(A_z\) rānei i te 3D).
– Kei te Vector B ngā wāhanga \(B_x\) me \(B_y\) i te 2D (me \(B_z\) rānei i te 3D).
Mā te tāpiri i ēnei whākapū e rua ka puta he whākapū hua R, ā, ko ēnei ngā wāhanga e whai ake nei:
\[ R_x = A_x + B_x \]
\[ R_y = A_y + B_y \]
Mō ngā whārite i roto i te 3D, ko te wāhanga z anō hoki e whai ake nei:
\[ R_z = A_z + B_z \]
I muri i te tatau i ia wāhanga o te whārite hua, ka kitea te modulus (rahi) me te ahunga o te whārite hua mā te whakamahi i te tātai:
\[ |R| = \sqrt{R_x^2 + R_y^2} \] (mō te 2D)
Mō te 3D rānei:
\[ |R| = \sqrt{R_x^2 + R_y^2 + R_z^2} \]
Ā, ka taea te whakatau i te ahunga o te ira hua mā te koki ki ngā tuaka taunga.
Ngā Pātai Tauira me te Kōrero
Pātai 1
E rua ngā whārite i roto i tētahi papa rua-ahu e hoatu ana:
– Kei te rawhiti a A \(5 \, \text{unit}\).
– Kei te raki a B \(3 \, \text{unit}\).
Whakatauhia te hua o te whārite R.
Kōrero
Tuatahi, ka hurihia e tātou te wetereo ki ōna wāhanga.
– Te Wētera A: \(A = (5, 0)\) nā te mea he wāhanga x anake tōna.
– Te Wēra B : \(B = (0, 3)\) nā te mea he wāhanga y anake tōna.
Anei te tapeke o ngā wāhanga:
\[ R_x = A_x + B_x = 5 + 0 = 5 \]
\[ R_y = A_y + B_y = 0 + 3 = 3 \]
Kātahi ka puta te hua o te whārite R:
\[ R = (5, 3) \]
Hei tatau i te roa (modulus) o te whārite R:
\[ |R| = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \tata ki te 5.83 \]
Ka taea te tatau i te ahunga o te whārite R mā te whakamahi i te koki θ ki te tuaka-x:
\[ \tan(\theta) = \frac{R_y}{R_x} = \frac{3}{5} \]
\[ \theta = \arctan\left(\frac{3}{5}\right) \approx 30.96^\circ \]
Nō reira, ko te roa o te whārite hua R he tata ki te 5.83 waeine, ā, he 30.96° te koki e hanga ana ki te tuaka-x.
Pātai 2
E rua ngā whārite i roto i ngā ahu-toru e hoatu ana:
– Ko A ko \(3\hat{i} + 2\hat{j} + 1\hat{k}\)
– Ko B ko \(1\hat{i} + 4\hat{j} + 2\hat{k}\)
Whakatauhia te hua o te whārite R.
Kōrero
Tuatahi, ka tautuhia e mātou ngā wāhanga o ia whārite:
– Te Wāhanga A : \(A_x = 3\), \(A_y = 2\), \(A_z = 1\).
– Te Wāhanga B : \(B_x = 1\), \(B_y = 4\), \(B_z = 2\).
Anei te tapeke o ngā wāhanga:
\[ R_x = A_x + B_x = 3 + 1 = 4 \]
\[ R_y = A_y + B_y = 2 + 4 = 6 \]
\[ R_z = A_z + B_z = 1 + 2 = 3 \]
Kātahi ka puta te hua o te whārite R:
\[ R = (4, 6, 3) \]
Hei tatau i te roa (modulus) o te whārite R:
\[ |R| = \sqrt{4^2 + 6^2 + 3^2} = \sqrt{16 + 36 + 9} = \sqrt{61} \tata ki te 7.81 \]
Ka taea te tatau i te ahunga o te whārite R e pā ana ki ngā tuaka x, y, me z mā te whakamahi i te cosine o te kaiwhakahaere:
\[ \cos(\alpha) = \frac{R_x}{|R|} = \frac{4}{7.81} \tata ki te 0.512 \]
\[ \alpha = \arccos(0.512) \tata ki te 59.50^\circ \]
\[ \cos(\beta) = \frac{R_y}{|R|} = \frac{6}{7.81} \tata ki te 0.768 \]
\[ \beta = \arccos(0.768) \tata ki te 39.50^\circ \]
\[ \cos(\gamma) = \frac{R_z}{|R|} = \frac{3}{7.81} \tata ki te 0.384 \]
\[ \gamma = \arccos(0.384) \tata ki te 67.64^\circ \]
Nō reira, ko te roa o te whārite hua R he tata ki te 7.81 ngā waeine, ā, ko ōna ahunga e pā ana ki ngā tuaka x, y, me z he 59.50°, 39.50°, me te 67.64°.
Pātai 3
E rua ngā whārite e hoatu ana:
– E 4 ngā waeine o te rahi o P, ā, e 45° te koki o tōna koki ki te tuaka-x pai.
– E 6 ngā waeine o te rahi o Q, ā, e 120° te koki o tōna koki ki te tuaka-x pai.
Whakatauhia te hua o te whārite R.
Kōrero
Tuatahi, ka wehea e tātou te whārite ki ōna wāhanga x me y:
– Te Wēka P : \(P_x = 4\cos(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} \approx 2.83\), \(P_y = 4\sin(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} \approx 2.83\).
– Te Wheketere Q : \(Q_x = 6\cos(120^\circ) = 6 \cdot \left(-\frac{1}{2}\right) = -3\), \(Q_y = 6\sin(120^\circ) = 6 \cdot \frac{\sqrt{3}}{2} \approx 5.2\).
Anei te tapeke o ngā wāhanga:
\[ R_x = P_x + Q_x = 2.83 – 3 = -0.17 \]
\[ R_y = P_y + Q_y = 2.83 + 5.2 = 8.03 \]
Kātahi, ko te hua o te whārite R ko:
\[ R = (-0.17, 8.03) \]
Hei tatau i te roa (modulus) o te whārite R:
\[ |R| = \sqrt{(-0.17)^2 + 8.03^2} = \sqrt{0.0289 + 64.48} = \sqrt{64.509} \tata ki te 8.03 \]
Te ahunga o te whārite R:
\[ \tan(\theta) = \frac{R_y}{R_x} = \frac{8.03}{-0.17} = -47.24 \]
\[ \theta = \arctan(-47.24) \approx -88.99^\circ \]
Heoi, ka inehia tēnei koki e pā ana ki te tuaka-x kino, nō reira ko te koki tuturu i roto i te horopaki o te raruraru ko:
\[ 180^\circ – 88.99^\circ \tata ki te 91.01^\circ \]
Nō reira, ko te roa o te whārite hua R he tata ki te 8.03 waeine, ā, ka hanga he koki o te 91.01° me te tuaka-x pai.
Kua matapakihia e tēnei tuhinga te tāpiritanga whārite ā-wāhanga, me te whakarato i ētahi tauira rapanga me ngā otinga. He tino whai hua te tikanga ā-wāhanga ki te whakahaere i ngā tataunga me te whakarato i tētahi huarahi pūnaha hei whakaoti rapanga whārite i roto i te taha pāngarau o te wāhi.
