Ngā Tauira Pātai e Matapaki ana i ngā Wētera Ahu-Rua i roto i tētahi Pūnaha Taunga
He rahinga kei a ia te rahi me te ahunga o te whārite. He maha ngā whakamahinga o ngā whārite i roto i ngā kaupapa pāngarau me te ahupūngao hei tohu i ngā āhuatanga rerekē. I roto i tēnei tuhinga, ka matapakihia e mātou ngā tauira o ngā whārite rua-ahu i roto i tētahi pūnaha taunga.
Ngā Ariā Taketake o ngā Wekita i roto i ngā Pūnaha Taunga
Ka taea te whakaatu i tētahi whārite i roto i tētahi pūnaha taunga rua-ahu pēnei \(\vec{A} = (a_1, a_2)\), ko \(a_1\) te wāhanga-x o te whārite, ā, ko \(a_2\) te wāhanga-y o te whārite. Ka taea te wehewehe i te whārite kia rua ngā wāhanga, arā, ko te wāhanga-x me te wāhanga-y.
Te Tāpiri me te Tangohanga o te Wetere
Ko te tāpiritanga o ngā whārite e rua \(\vec{A} = (a_1, a_2)\) me \(\vec{B} = (b_1, b_2)\) ko:
\[
\vec{A} + \vec{B} = (a_1 + b_1, a_2 + b_2)
\]
Ahakoa ko te whakaiti:
\[
\vec{A} – \vec{B} = (a_1 – b_1, a_2 – b_2)
\]
Whakareatanga Tauine
Mena ko \(\vec{A} = (a_1, a_2)\) ā, he tauine tauine a \(k\), ko \(k\vec{A}\) tēnei:
\[
k\vec{A} = (k \cdot a_1, k \cdot a_2)
\]
Te Rahi o te Wēka
Ko te rahi, te roa rānei o te whārite \(\vec{A} = (a_1, a_2)\) ko:
\[
|\vec{A}| = \sqrt{a_1^2 + a_2^2}
\]
Wāhanga Wāhanga
Ko te ira kotahi he ira kotahi te roa. Ko te ira kotahi o \(\vec{A} = (a_1, a_2)\) ko:
\[
\hat{A} = \frac{\vec{A}}{|\vec{A}|} = \left( \frac{a_1}{\sqrt{a_1^2 + a_2^2}}, \frac{a_2}{\sqrt{a_1^2 + a_2^2}} \right)
\]
Ngā Pātai Tauira me te Kōrero
Pātai 1: Te Tāpiri me te Tango i ngā Wēka
E rua ngā whārite e hoatu ana penei: \(\vec{A} = (3, 4)\) me \(\vec{B} = (1, 2)\). Kimihia te hua o \(\vec{A} + \vec{B}\) me \(\vec{A} – \vec{B}\).
Kōrero:
\[
\vec{A} + \vec{B} = (3 + 1, 4 + 2) = (4, 6)
\]
\[
\vec{A} – \vec{B} = (3 – 1, 4 – 2) = (2, 2)
\]
Pātai 2: Whakarea Tauine
I runga i te whārite \(\vec{C} = (2, -3)\), tatauhia \(3\vec{C}\) me \(-2\vec{C}\).
Kōrero:
\[
3\vec{C} = 3 \cdot (2, -3) = (6, -9)
\]
\[
-2\vec{C} = -2 \cdot (2, -3) = (-4, 6)
\]
Pātai 3: Te Rahi o te Wētera
Tātaihia te rahi o te whārite \(\vec{D} = (5, 12)\).
Kōrero:
\[
|\vec{D}| = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]
Pātai 4: Ngā Wētera Wāhanga
Kimihia te ine kotahi o te ine \(\vec{E} = (4, 3)\).
Kōrero:
\[
|\vec{E}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]
\[
\hat{E} = \frac{\vec{E}}{|\vec{E}|} = \left( \frac{4}{5}, \frac{3}{5} \right)
\]
Pātai 5: Te Tūnga me te Tawhiti o ngā Waehere
E rua ngā pūwāhi i roto i te papa taunga rua-ahu ko P(2, 3) me Q(5, 7). Tāutuhia te ira tūnga mai i te pūwāhi P ki te pūwāhi Q me te tawhiti i waenganui i a rāua.
Kōrero:
Ko te tūnga o te weketere mai i P ki Q ko:
\[
\vec{PQ} = \vec{Q} – \vec{P} = (5 – 2, 7 – 3) = (3, 4)
\]
Ko te tawhiti i waenganui i ngā pūwāhi P me Q ko:
\[
|\vec{PQ}| = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
Pātai 6: Hua o te Hua Ira
Mena ko \(\vec{F} = (-3, 4)\) me \(\vec{G} = (2, 1)\), tatauhia te hua ira o \(\vec{F} \cdot \vec{G}\).
Kōrero:
Ko te hua ira o ngā pūwero e rua ko:
\[
\vec{F} \cdot \vec{G} = (-3) \cdot 2 + 4 \cdot 1 = -6 + 4 = -2
\]
Pātai 7: Te Koki i waenganui i ngā Wētera e Rua
Mena ko \(\vec{H} = (7, -4)\) me \(\vec{I} = (3, 0)\), whakatauhia te koki i waenganui i ngā whārite e rua.
Kōrero:
Hei whakatau i te koki i waenganui i ngā whārite e rua, ka whakamahia e mātou te tātai:
\[
\cos \theta = \frac{\vec{H} \cdot \vec{I}}{|\vec{H}| |\vec{I}|}
\]
Tuatahi, tatauhia te hua ira \(\vec{H} \cdot \vec{I}\):
\[
\vec{H} \cdot \vec{I} = 7 \cdot 3 + (-4) \cdot 0 = 21 + 0 = 21
\]
Kātahi, tatauhia ngā rahi o \(\vec{H}\) me \(\vec{I}\):
\[
|\vec{H}| = \sqrt{7^2 + (-4)^2} = \sqrt{49 + 16} = \sqrt{65}
\]
\[
|\vec{I}| = \sqrt{3^2 + 0^2} = \sqrt{9} = 3
\]
Whakauruhia ēnei uara ki roto i te tātai:
\[
\cos \theta = \frac{21}{\sqrt{65} \cdot 3} = \frac{21}{3\sqrt{65}} = \frac{7}{\sqrt{65}}
\]
Nō reira, \(\theta = \cos^{-1} \left( \frac{7}{\sqrt{65}} \right) \).
Pātai 8: Whakaaturanga Wetereo
Mō ngā whārite \(\vec{J} = (2, 1)\) me \(\vec{K} = (-1, 3)\), tatauhia te matapae o \(\vec{J}\) ki runga i \(\vec{K}\).
Kōrero:
Ko te matapae o \(\vec{J}\) i runga i \(\vec{K}\) ko:
\[
\text{proj}_{\vec{K}} \vec{J} = \maui( \frac{\vec{J} \cdot \vec{K}}{|\vec{K}|^2} \matau) \vec{K}
\]
Tuatahi, tatauhia te hua ira \(\vec{J} \cdot \vec{K}\):
\[
\vec{J} \cdot \vec{K} = 2 \cdot (-1) + 1 \cdot 3 = -2 + 3 = 1
\]
Kātahi, ko te rahi o \(\vec{K}\):
\[
|\vec{K}| = \sqrt{(-1)^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}
\]
Nō reira,:
\[
|\vec{K}|^2 = 10
\]
Tomo ki roto i te tātai:
\[
\text{proj}_{\vec{K}} \vec{J} = \left( \frac{1}{10} \right) \vec{K} = \left( \frac{1}{10} \right) (-1, 3) = \left( -\frac{1}{10}, \frac{3}{10} \right)
\]
Koinei ngā tauira o ngā raruraru me ngā kōrero e pā ana ki ngā whārite rua-ahu i roto i tētahi pūnaha taunga. He āwhina te mārama pai ki ngā whārite i roto i ngā tono maha i roto i te pāngarau, te ahupūngao, me te hangarau. Mā te whakaharatau ki ngā tauira maha ka taea te hōhonu ake i tō māramatanga ki tēnei ariā, ka taea ai te whakamahi pai i roto i ngā āhuatanga rerekē.
