Ngā Tauira Pātai e Matapaki ana i ngā Wetere me ngā Pūnaha Taunga
Ehara i te mea ko ngā tau me ngā tātai uaua anake te pāngarau; engari ko te mārama ki ngā ariā taketake e hanga ana i te tūāpapa o ngā tono o te ao tūturu. Ko tētahi o ngā ariā tino nui i roto i te pāngarau ko ngā whārite me ngā pūnaha taunga. I roto i tēnei tuhinga, ka tūhuratia e tātou ngā tauira raruraru me te matapaki i ngā whārite me ngā pūnaha taunga hei whakawhānui ake i tō tātou māramatanga ki te kaupapa.
He Kupu Whakataki ki ngā Wetereo
I mua i te ruku ki ngā tauira me ngā kōrero, he mea nui kia mārama ki ngā kaupapa matua o ngā whārite me ngā pūnaha taunga. Ko te whārite he mea he rahi, he ahunga hoki tōna. Ka taea te whakaatu i ngā whārite i roto i ngā whānuitanga rerekē, engari i roto i tēnei tuhinga, ka arotahi tātou ki ngā whārite rua-ahu (2D).
Ko te tikanga, ka tuhia te wetere i ngā taha e rua i te āhua nei:
\[ \vec{v} = \begin{pmatrix} x \\ y \end{pmatrix} \]
Ko \(x\) me \(y\) ngā wāhanga o te ira i roto i ngā taunga x me y.
Pūnaha Taunga Kāretiana
Ko te pūnaha taunga Cartesian te pūnaha taunga e whakamahia whānuitia ana i roto i te pāngarau. E rua ngā rārangi poutū e whakamahia ana, te tuaka-x me te tuaka-y, hei whakatau i te tūranga o tētahi pūwāhi i runga i tētahi papa. Ko ngā pūwāhi \( (x, y) \) e tohu ana i ngā tūranga whakapae me te poutū o tētahi pūwāhi e pā ana ki te pūtake (0,0).
Ngā Pātai Tauira me te Kōrero
Nā, ka tirohia e tātou ētahi tauira rapanga e pā ana ki ngā whārite me ngā pūnaha taunga.
Tauira Pātai 1: Tāpiritanga Wetere
Pātai: E rua ngā whārite \( \vec{a} \) me \( \vec{b} \) e hoatu ana penei:
\[ \vec{a} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} \]
\[ \vec{b} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \]
Tātaihia te hua o te tāpiritanga \( \vec{a} + \vec{b} \).
Kōrero:
Ka tāpirihia ngā whārite e rua mā te tāpiri i ngā wāhanga e rite ana. Nō reira,
\[ \vec{a} + \vec{b} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} + \begin{pmatrix} 1 \\ 2 \end{pmatrix} \]
Te tukanga tāpiri:
\[ \vec{a} + \vec{b} = \begin{pmatrix} 3 + 1 \\ 4 + 2 \end{pmatrix} \]
Ko te hua:
\[ \vec{a} + \vec{b} = \begin{pmatrix} 4 \\ 6 \end{pmatrix} \]
Nō reira, ko te hua o te tāpiri i ngā whārite \( \vec{a} \) me \( \vec{b} \) ko \( \begin{pmatrix} 4 \\ 6 \end{pmatrix} \).
Tauira Pātai 2: Te Tangohanga Wetereo
Pātai: E rua ngā whārite \( \vec{a} \) me \( \vec{c} \) e hoatu ana penei:
\[ \vec{a} = \begin{pmatrix} 5 \\ 7 \end{pmatrix} \]
\[ \vec{c} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} \]
Tātaihia te hua o te tangohanga \( \vec{a} – \vec{c} \).
Kōrero:
Ka tangohia ngā wāhanga e rite ana mā te tango i ngā whārite e rua. Nō reira,
\[ \vec{a} – \vec{c} = \begin{pmatrix} 5 \\ 7 \end{pmatrix} – \begin{pmatrix} 2 \\ 3 \end{pmatrix} \]
Te tukanga whakaiti:
\[ \vec{a} – \vec{c} = \begin{pmatrix} 5 – 2 \\ 7 – 3 \end{pmatrix} \]
Ko te hua:
\[ \vec{a} – \vec{c} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} \]
Nō reira, ko te hua o te tango i te whārite \( \vec{a} \) mai i \( \vec{c} \) ko \( \begin{pmatrix} 3 \\ 4 \end{pmatrix} \).
Tauira 3: Te Rahi o te Wētera
Pātai: I hoatu he whārite \( \vec{d} \):
\[ \vec{d} = \begin{pmatrix} 6 \\ 8 \end{pmatrix} \]
Tātaihia te rahi o te whārite \( \vec{d} \).
Kōrero:
Ka tatauhia te rahi o tētahi whārite \( \vec{d} = \begin{pmatrix} x \\ y \end{pmatrix} \) mā te tātai:
\[ \| \vec{d} \| = \sqrt{x^2 + y^2} \]
Mō te whārite \( \vec{d} \):
\[ \| \vec{d} \| = \sqrt{6^2 + 8^2} \]
Te tukanga tatau:
\[ \| \vec{d} \| = \sqrt{36 + 64} \]
\[ \| \vec{d} \| = \sqrt{100} \]
\[ \| \vec{d} \| = 10 \]
Nō reira, ko te rahi o te whārite \( \vec{d} \) he 10.
Tauira Pātai 4: Ngā Taunga Waenganui
Pātai: Homai te pūwāhi A (2,3) me te pūwāhi B (8,7). Whakatauhia ngā taunga o waenganui o te rārangi e hono ana i ngā pūwāhi A me B.
Kōrero:
Ka taea te tatau i ngā taunga o waenganui o te rārangi e hono ana i ngā pūwāhi e rua \( A \) me \( B \) mā te whakamahi i te tātai:
\[ \text{Pūwāhi Waenga} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Whakakapia ngā taunga o ngā pūwāhi A me B:
\[ \text{Pūwāhi Waenga} = \left( \frac{2 + 8}{2}, \frac{3 + 7}{2} \right) \]
Te tukanga tatau:
\[ \text{Pūwāhi Waenga} = \left( \frac{10}{2}, \frac{10}{2} \right) \]
\[ \text{Waenganui} = (5, 5) \]
Nō reira, ko ngā taunga o waenganui o te rārangi e hono ana i ngā pūwāhi A me B ko (5,5).
Tauira Pātai 5: Whakarea Tauine mā te Wetere
Pātai: I hoatu he whārite \( \vec{e} \):
\[ \vec{e} = \begin{pmatrix} 4 \\ 3 \end{pmatrix} \]
Whakareatia te ira \( \vec{e} \) ki te tauine 2.
Kōrero:
Ko te whakarea o tētahi ira ki tētahi tauine ka mahia mā te whakarea i ia wāhanga o te ira ki te tauine. Nō reira,
\[ 2 \times \vec{e} = 2 \times \begin{pmatrix} 4 \\ 3 \end{pmatrix} \]
Te tukanga whakarea:
\[ 2 \times \vec{e} = \begin{pmatrix} 2 \times 4 \\ 2 \times 3 \end{pmatrix} \]
Ko te hua:
\[ 2 \times \vec{e} = \begin{pmatrix} 8 \\ 6 \end{pmatrix} \]
Nō reira, ko te hua o te whakarea i te ira \( \vec{e} \) ki te tauine 2 ko \( \begin{pmatrix} 8 \\ 6 \end{pmatrix} \).
Whakamutunga
I roto i te pāngarau, he mea nui ngā ariā o ngā whārite me ngā pūnaha taunga ki te mārama ki ngā āhuatanga rerekē, i roto i te ariā me ā rātou tono puta noa i ngā mara rerekē. Mā te mārama ki ngā mahi whārite taketake pēnei i te tāpiri, te tango, te whakarea tauine, me ngā tataunga rahi, me te tono o ngā pūnaha taunga, ka taea e tātou te mārama ake ki ngā raruraru uaua ake.
Ko te whakaharatau tonu te mea nui hei matatau i ēnei ariā. He tīmatanga pai ngā tauira rapanga i runga ake nei hei whakahōhonu ake i tō māramatanga ki ngā whārite me ngā pūnaha taunga. Whakamātauria ētahi atu rapanga, kia kitea ai te ataahua o te pāngarau mā te tūhura atu.
