Ngā Tauira Pātai me te Kōrero mō ngā Wāhanga Wetere
He ariā taketake ngā whārite i roto i te ahupūngao me te pāngarau, e whakamahia ana hei whakaahua i ngā rahinga me te rahi me te ahunga. He mea nui te māramatanga hōhonu ki ngā whārite hei whakaoti rapanga i roto i te pūtaiao me te hangarau. Ka matapakihia e tēnei tuhinga ētahi tauira rapanga e pā ana ki ngā wāhanga whārite, me ō rātou whakamārama.
He Kupu Whakataki ki ngā Wetereo
He rahinga te whārite e rua ōna āhuatanga matua: te rahi me te ahunga. Hei tauira, he rahinga whārite te tere nā te mea kei a ia te rahi (te tere) me te ahunga (te wāhi e haere ana). Hei tohu i ngā whārite, he maha ā tātou whakamahi i ngā pere, ko te roa o te pere e tohu ana i tōna rahi, ā, ko te ahunga o te pere e tohu ana i tōna ahunga.
He maha ngā wā ka whakaatuhia he ira i roto i te wāhi rua-ahu ko 𝐀 = 𝑎ᵢ + 𝑏ⱼ, ko 𝑎 me 𝑏 ngā wāhanga o te ira i runga i ngā tuaka-x me y, ā, ko 𝐢 me 𝐣 ngā ira wae i runga i ngā tuaka-x me y.
Tauira Pātai 1: Te Whakatau i ngā Wāhanga Wetereo mai i tētahi Whakaaturanga Whakairoiro
Pātai: He tīmatanga kei te pūtake (0,0) me te mutunga kei ngā taunga (4,3) o te whārite 𝐀. Whakatauhia ngā wāhanga o te whārite 𝐀.
Kōrero: Ka taea te tuhi i te wetere e tīmata ana mai i te pūwāhi tīmatanga (0,0) ki te pūwāhi whakamutunga (4,3) ki te āhua wāhanga pēnei i te 𝐀 = 4𝐢 + 3𝐣. Ko te wāhanga i te tuaka-x he 4, ā, i te tuaka-y he 3.
Tauira Pātai 2: Te Whakatau i te Rahi o te Wetere
Rapanga: Tātaihia te rahi o te whārite 𝐀 = 4𝐢 + 3𝐣.
Kōrero: Ka taea te tatau i te rahi (te rahi rānei) o tētahi whārite 𝐀 mā te whakamahi i te tātai Pythagoras, arā:
\[ |𝐀| = \sqrt{𝑎² + 𝑏²} \]
Mō te whārite 𝐀 = 4𝐢 + 3𝐣, kātahi:
\[ |𝐀| = \sqrt{4² + 3²} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
Nō reira, ko te rahi o te whārite 𝐀 he 5 waeine.
Tauira 3: Te Tāpiri i ngā Wētera e Rua
Pātai: E rua ngā whārite 𝐁 = 2𝐢 + 3𝐣 me 𝐂 = -𝐢 + 4𝐣. Tātaihia te tapeke o ngā whārite 𝐁 me 𝐂.
Kōrero: Hei tāpiri i ngā whārite e rua, ka tāpirihia noa ngā wāhanga i te ahunga kotahi o ia whārite:
\[ 𝐁 + 𝐂 = (2𝐢 + 3𝐣) + (-𝐢 + 4𝐣) \]
\[ = (2 + (-1))𝐢 + (3 + 4)𝐣 \]
\[ = 1𝐢 + 7𝐣 \]
Nō reira, ko te hua o te tāpiri i ngā whārite 𝐁 me 𝐂 ko 𝐃 = 𝐢 + 7𝐣.
Tauira Pātai 4: Te Tātai i te Koki i waenganui i ngā Wētera e Rua
Rapanga: E rua ngā whārite e hoatu ana, arā, 𝐀 = 3𝐢 + 4𝐣 me 𝐁 = 4𝐢 – 3𝐣. Tātaihia te koki i waenganui i ngā whārite e rua.
Kōrero: Ka taea te tatau i te koki i waenganui i ngā whārite e rua mā te whakamahi i te tātai cosine:
\[ \cos(𝜃) = \frac{𝐀 · 𝐁}{|𝐀| |𝐁|} \]
1. Tātaihia te hua ira (𝐀 · 𝐁):
\[ 𝐀 · 𝐁 = (3𝐢 + 4𝐣) · (4𝐢 – 3𝐣) \]
\[ = (3 4) + (4 -3) \]
\[ = 12 – 12 \]
\[ = 0 \]
2. Tātaihia te rahi o ngā whārite 𝐀 me 𝐁:
\[ |𝐀| = \sqrt{3² + 4²} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
\[ |𝐁| = \sqrt{4² + (-3)²} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
3. Tāpirihia ki te tātai kosinī:
\[ \cos(𝜃) = \frac{0}{5 5} = 0 \]
Nā te mea ko te 0 te 0, ko te 90° te 0. Nō reira, ko te koki i waenganui i ngā whārite e rua he 90 nekehanga.
Tauira Pātai 5: Te Tatau i te Hua Whakawhiti o ngā Wetere
Raru: Ki te hoatu he whārite e rua i roto i ngā ahu-toru, 𝐀 = 𝐢 + 2𝐣 + 3𝐤 me 𝐁 = 4𝐢 + 5𝐣 + 6𝐤, tatauhia te whārite whakawhiti 𝐀 × 𝐁.
Kōrero: Ko te hua whakawhiti o ngā whārite e rua i roto i ngā ahu-toru (𝐀 × 𝐁) ko:
\[ 𝐀 × 𝐁 = \begin{vmatrix} 𝐢 & 𝐣 & 𝐤 \\ 1 & 2 & 3 \\ 4 & 5 & 6 \end{vmatrix} \]
\[ = 𝐢 (2 6 – 3 5) – 𝐣 (1 6 – 3 4) + 𝐤 (1 5 – 2 4) \]
\[ = 𝐢 (12 – 15) – 𝐣 (6 – 12) + 𝐤 (5 – 8) \]
\[ = 𝐢 (-3) – 𝐣 (-6) + 𝐤 (-3) \]
\[ = -3𝐢 + 6𝐣 – 3𝐤 \]
Nō reira, ko te hua o te hua whakawhiti 𝐀 × 𝐁 ko -3𝐢 + 6𝐣 – 3𝐤.
Whakamutunga
I roto i te ahupūngao me te pāngarau, he huarahi tino whai hua ngā whārite hei whakaatu i ngā rahinga he ahunga me te rahi. Mā te mārama ki te whakatau i ngā wāhanga whārite, te tatau i ngā rahi, te tāpiri i ngā whārite, me te tatau i ngā koki i waenga i ngā whārite me ngā hua whakawhiti, ka taea e tātou te whakaoti rapanga maha e pā ana ki ngā whārite. Ko te matapaki i ngā tauira rapanga i runga ake nei he āwhina i te hōhonu ake o tō tātou māramatanga ki tēnei ariā. Hei whakamutunga, ko te kaha ki te mārama me te whakahaere me ngā whārite he pūkenga tino whai hua i roto i ngā momo mara o te pūtaiao me te hangarau.
