Tauira o ngā Pātai Kōrero mō te Mahi Wetereo
He ariā taketake ngā mahi taupū i roto i te pāngarau e puta pinepine ana i roto i ngā momo mara rangahau, pērā i te ahupūngao, te hangarau, me te pūtaiao rorohiko. I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira o ngā mahi taupū me ā rātou otinga hei whakarato i tētahi māramatanga hohonu ake, tūturu ake hoki. Ka hipokina e ēnei tauira ngā mahi taketake pērā i te tāpiri me te tango taupū, tae atu ki ngā mahi matatau ake pērā i te whakarea tauine me te whakarea whakawhiti taupū.
1. Te Tāpiri me te Tangohanga o te Wētera
Tauira Pātai 1
E rua ngā whārite A me B i te āhua wāhanga:
\[ \mathbf{A} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} \]
\[ \mathbf{B} = \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} \]
Tātaihia te hua o te tāpiri me te tango o ngā whārite e rua.
Kōrero
Mō te tāpiritanga whārite, ka tāpirihia e mātou ia wāhanga e rite ana o ngā whārite e rua.
\[ \mathbf{A} + \mathbf{B} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} + \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 + (-1) \\ 3 + 4 \\ -1 + 2 \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \\ 1 \end{pmatrix} \]
Mō te tango whārite, ka tangohia e tātou ia wāhanga e rite ana o ngā whārite e rua.
\[ \mathbf{A} – \mathbf{B} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} – \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 – (-1) \\ 3 – 4 \\ -1 – 2 \end{pmatrix} = \begin{pmatrix} 3 \\ -1 \\ -3 \end{pmatrix} \]
2. Te Whakarea Tauine mā te Wetere
Tauira Pātai 2
I hoatu he whārite C me te tauine k:
\[ \mathbf{C} = \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix} \]
\[ k = 4 \]
Tātaihia te hua tauine o te whārite C mā te tauine k.
Kōrero
Ka whakareatia te tauine ira ki te ira whārite mā te whakarea i ia wāhanga o te ira whārite ki te tauine whārite.
\[ k \mathbf{C} = 4 \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix} = \begin{pmatrix} 4 \cdot 1 \\ 4 \cdot (-2) \\ 4 \cdot 3 \end{pmatrix} = \begin{pmatrix} 4 \\ -8 \\ 12 \end{pmatrix} \]
3. Hua Ira
Tauira Pātai 3
E rua ngā whārite D me E i hoatu:
\[ \mathbf{D} = \begin{pmatrix} 3 \\ -2 \\ 4 \end{pmatrix} \]
\[ \mathbf{E} = \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix} \]
Tātaihia te hua ira o ngā whārite e rua.
Kōrero
Ka whiwhihia te hua ira o ngā whārite e rua mā te tāpiri i ngā hua o ō rāua wāhanga e rite ana.
\[ \mathbf{D} \cdot \mathbf{E} = 3 \cdot 1 + (-2) \cdot 0 + 4 \cdot (-1) = 3 + 0 – 4 = -1 \]
4. Hua Whakawhiti
Tauira Pātai 4
E rua ngā whārite F me G i hoatu:
\[ \mathbf{F} = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} \]
\[ \mathbf{G} = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix} \]
Tātaihia te hua whakawhiti o ngā whārite e rua.
Kōrero
Ka whiwhihia te hua whakawhiti o ngā whārite e rua i roto i te wāhi toru-ahu mā te whakamahi i te whakatau o te matihiko i hangaia e aua whārite. Ka hoatuhia te hua whakawhiti e te tātai:
\[ \mathbf{F} \times \mathbf{G} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 3 & 4 \\ 1 & -1 & 2 \end{vmatrix} \]
Ka taea te tatau i tēnei penei:
\[
\mathbf{F} \times \mathbf{G} = \mathbf{i} \begin{vmatrix} 3 & 4 \\ -1 & 2 \end{vmatrix} – \mathbf{j} \begin{vmatrix} 2 & 4 \\ 1 & 2 \end{vmatrix} + \v} 3 & -1 \mutu{vmatrix}
\]
Te tatau i te whakatau o ia whārite iti:
\[
= \mathbf{i} (3 \cdot 2 – 4 \cdot -1) – \mathbf{j} (2 \cdot 2 – 4 \cdot 1) + \mathbf{k} (2 \cdot -1 – 3 \cdot 1)
\]
\[
= \mathbf{i} (6 + 4) – \mathbf{j} (4 – 4) + \mathbf{k} (-2 – 3)
\]
\[
= \mathbf{i} (10) – \mathbf{j} (0) + \mathbf{k} (-5)
\]
\[
= \begin{pmatrix} 10 \\ 0 \\ -5 \end{pmatrix}
\]
Nā, ko te hua whakawhiti o F me G ko:
\[ \mathbf{F} \times \mathbf{G} = \begin{pmatrix} 10 \\ 0 \\ -5 \end{pmatrix} \]
5. Te whakatau i te koki i waenganui i ngā whārite e rua
Tauira Pātai 5
E rua ngā whārite H me I i hoatu:
\[ \mathbf{H} = \begin{pmatrix} 6 \\ 2 \\ 3 \end{pmatrix} \]
\[ \mathbf{I} = \begin{pmatrix} 1 \\ 4 \\ -2 \end{pmatrix} \]
Tāutuhia te koki i waenganui i ngā whārite e rua.
Kōrero
Ka kitea te koki \(\theta\) i waenganui i ngā whārite e rua mā te whakamahi i te whanaungatanga i waenganui i te hua ira me te rahi o ngā whārite e rua:
\[ \mathbf{H} \cdot \mathbf{I} = \| \mathbf{H} \| \| \mathbf{I} \| \cos \theta \]
Tuatahi, tatauhia te hua ira \( \mathbf{H} \cdot \mathbf{I} \):
\[ \mathbf{H} \cdot \mathbf{I} = 6 \cdot 1 + 2 \cdot 4 + 3 \cdot (-2) = 6 + 8 – 6 = 8 \]
Muri iho, tatauhia te rahi o ngā whārite e rua:
\[ \| \mathbf{H} \| = \sqrt{6^2 + 2^2 + 3^2} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]
\[ \| \mathbf{I} \| = \sqrt{1^2 + 4^2 + (-2)^2} = \sqrt{1 + 16 + 4} = \sqrt{21} \]
Kātahi ka whakakapia ēnei uara ki te tātai koki:
\[ \cos \theta = \frac{\mathbf{H} \cdot \mathbf{I}}{\| \mathbf{H} \| \| \mathbf{I} \|} = \frac{8}{7\sqrt{21}} \]
\[ \theta = \cos^{-1} \left( \frac{8}{7\sqrt{21}} \right) \]
Hei whakamutunga, ka taea e tātou te whakamahi i te tātaitai hei kimi i te uara o te koki:
\[ \theta \tata ki te 73,4^\circ \]
Whakamutunga
He mea nui te ariā o ngā mahi whārite i roto i te pāngarau me te pūtaiao. E matapakihia ana e tēnei tuhinga ētahi tauira rapanga me ō rātou otinga, mai i te tāpiri me te tango whārite, te whakarea tauine, te hua ira, te hua whakawhiti, me te whakatau i te koki i waenganui i ngā whārite e rua. Mā te mahi i ēnei tauira, ko te tumanako ka whakarei ake i tō māramatanga ki ngā mahi whārite, ā, ka āwhina i a koe ki te whakaoti rapanga e pā ana ki ngā whārite i roto i ngā horopaki rerekē.
