Tauira o tētahi Pātai Kōrero mō te Wētera Wāhanga o tētahi Wētera
Pendahuluan
I roto i te pāngarau me te ahupūngao, he huānga taketake ngā whārite e tohu ana i te rahi me te ahunga. He maha ngā wā ka whakamahia ngā whārite hei whakaahua i ngā āhuatanga rerekē pēnei i te tere, te kaha, me te nekehanga i roto i te wāhi rua-ahu, toru-ahu rānei. Ko tētahi ariā nui e pā ana ki ngā whārite ko te whārite kotahi. Ka matapakihia e tēnei tuhinga te whakamāramatanga o te whārite kotahi, me pēhea te tatau, me te whakarato i ētahi tauira rapanga me ngā otinga.
Te Mārama ki ngā Wētera Waeine
Ko te ira kotahi he ira kotahi te rahi, ā, he rite tonu te ahunga ki te ira taketake. He maha ngā wā ka whakamahia ngā ira kotahi hei whakahaere i te tātaritanga nā te mea kotahi tonu te rahi, ā, ko te arotahi matua ko tōna ahunga. Hei huri i tētahi ira ki tētahi ira kotahi, me wehewehe ia wāhanga mā te rahi o te ira.
I roto i te pāngarau, mēnā he whārite te \( \mathbf{v} \), ka taea te whakaatu i tōna whārite waeine \( \mathbf{\hat{v}} \) penei:
\[
\mathbf{\hat{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}
\]
ko \( \|\mathbf{v}\| \) te rahi, te roa rānei o te whārite \( \mathbf{v} \).
Te Tātai i te Rahi o te Wētera
Ka taea te tatau i te rahi o tētahi ira \( \mathbf{v} \) i roto i te wāhi rua-ahu me ngā wāhanga \( (v_x, v_y) \) mā te whakamahi i te tātai:
\[
\|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2}
\]
I taua wā, mō ngā whārite i roto i te wāhi toru-ahu me ngā wāhanga \( (v_x, v_y, v_z) \), ka tatauhia te rahi mā te whakamahi i te tātai:
\[
\|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2 + v_z^2}
\]
Ngā Pātai Tauira me te Kōrero
Hei whakamārama i te ariā o ngā whārite wae, me titiro tātou ki ētahi tauira pātai me ā rātou matapakinga.
Tauira Pātai 1
Pātai: Hoatu he whārite \( \mathbf{a} = (3, 4) \). Whakatauhia te whārite kotahi o te whārite \( \mathbf{a} \).
Kōrero:
1. Whakatauhia ngā wāhanga o te whārite \( \mathbf{a} \):
\( a_x = 3 \), \( a_y = 4 \)
2. Tātaihia te rahi o te whārite \( \mathbf{a} \):
\[
\|\mathbf{a}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
3. Tātaihia te ira kotahi mā te wehewehe i ia wāhanga o te ira \( \mathbf{a} \) mā tōna rahi:
\[
\mathbf{\hat{a}} = \left( \frac{3}{5}, \frac{4}{5} \right) = \left( 0.6, 0.8 \right)
\]
Nō reira, ko te ine ine o \( \mathbf{a} \) ko \( (0.6, 0.8) \).
Tauira Pātai 2
Pātai: Hoatu he whārite \( \mathbf{b} = (1, -2, 2) \). Whakatauhia te whārite kotahi o te whārite \( \mathbf{b} \).
Kōrero:
1. Whakatauhia ngā wāhanga o te whārite \( \mathbf{b} \):
\( b_x = 1 \), \( b_y = -2 \), \( b_z = 2 \)
2. Tātaihia te rahi o te whārite \( \mathbf{b} \):
\[
\|\mathbf{b}\| = \sqrt{1^2 + (-2)^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3
\]
3. Tātaihia te ira kotahi mā te wehewehe i ia wāhanga o te ira \( \mathbf{b} \) mā tōna rahi:
\[
\mathbf{\hat{b}} = \left( \frac{1}{3}, \frac{-2}{3}, \frac{2}{3} \right) \approx \left( 0.333, -0.667, 0.667 \right)
\]
Nō reira, ko te ine ira o \( \mathbf{b} \) ko \( \left( 0.333, -0.667, 0.667 \right) \).
Tauira Pātai 3
Pātai: I hoatu te whārite \( \mathbf{c} = (-7, 24) \). Whakatauhia te whārite kotahi o te whārite \( \mathbf{c} \).
Kōrero:
1. Whakatauhia ngā wāhanga o te whārite \( \mathbf{c} \):
\( c_x = -7 \), \( c_y = 24 \)
2. Tātaihia te rahi o te whārite \( \mathbf{c} \):
\[
\|\mathbf{c}\| = \sqrt{(-7)^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25
\]
3. Tātaihia te ira kotahi mā te wehewehe i ia wāhanga o te ira \( \mathbf{c} \) mā tōna rahi:
\[
\mathbf{\hat{c}} = \left( \frac{-7}{25}, \frac{24}{25} \right) = \left( -0.28, 0.96 \right)
\]
Nō reira, ko te ine ine o \( \mathbf{c} \) ko \( (-0.28, 0.96) \).
Tauira Pātai 4
Pātai: Mēnā he whārite \( \mathbf{d} = (6, 8, 0) \), whakatauhia te whārite wae o te whārite \( \mathbf{d} \).
Kōrero:
1. Whakatauhia ngā wāhanga o te whārite \( \mathbf{d} \):
\( d_x = 6 \), \( d_y = 8 \), \( d_z = 0 \)
2. Tātaihia te rahi o te whārite \( \mathbf{d} \):
\[
\|\mathbf{d}\| = \sqrt{6^2 + 8^2 + 0^2} = \sqrt{36 + 64 + 0} = \sqrt{100} = 10
\]
3. Tātaihia te ira kotahi mā te wehewehe i ia wāhanga o te ira \( \mathbf{d} \) mā tōna rahi:
\[
\mathbf{\hat{d}} = \left( \frac{6}{10}, \frac{8}{10}, \frac{0}{10} \right) = \left( 0.6, 0.8, 0 \right)
\]
Nō reira, ko te ine ira o \( \mathbf{d} \) ko \( (0.6, 0.8, 0) \).
Te Katinga
Mā te kōrero me ngā tauira i runga ake nei, ka taea e tātou te mārama ko te tatau i tētahi ira waeine me tatau i te rahi o te ira, kātahi ka wehewehe i ngā wāhanga o te ira ki taua rahi. He tino whai hua ngā ira waeine i roto i ngā tono maha pērā i te whakataurite ira waeine i roto i ngā whakairoiro rorohiko, te tātari kaha i roto i te ahupūngao, me te maha atu o ngā mara. Mā te mārama ki tēnei ariā, me ngāwari ake tā tātou whakahaere i ngā raruraru e pā ana ki ngā ira waeine.
