Tauira o ngā Raru e Matapaki ana i ngā Wētera Whakamuri
He mea pāngarau te whārite, he rahi, he ahunga hoki tōna. I roto i te ako i ngā whārite, he maha ngā wā ka tūtaki tātou ki ngā whārite me ngā āhuatanga motuhake. Ko tētahi ariā matua o ngā whārite ko te whārite whakamuri, te whārite kino rānei. Ka kapi tēnei tuhinga i ngā tauira me te matapakinga o ngā whārite whakamuri.
Te Mārama ki ngā Wētera Whakamuri
Ko te whārite whakamuri, e kiia ana he whārite kino, he whārite he rite te rahi engari he rerekē te ahunga ki te whārite taketake. Mena ka tohua he whārite e \(\vec{a}\), ko tōna whārite whakamuri ko \(-\vec{a}\). Mā te pāngarau, mena \(\vec{a} = (a_1, a_2, a_3)\), ko \(-\vec{a} = (-a_1, -a_2, -a_3)\).
Tauira Pātai 1
I hoatu te whārite \(\vec{a} = (3, 4, -2)\). Whakatauhia te whārite whakamuri o \(\vec{a}\).
Kōrero:
Hei whakatau i te whārite whakamuri o \(\vec{a}\), me huri noa i ia wāhanga whārite ki te kino:
\[
-\vec{a} = (-3, -4, 2)
\]
Nō reira, ko te whārite whakamuri o \(\vec{a} = (3, 4, -2)\) ko \(-\vec{a} = (-3, -4, 2)\).
Tauira Pātai 2
Me waiho te whārite \(\vec{b} = (7, -5, 0)\). Kimihia te whārite whakamuri o \(\vec{b}\) ā, manatokohia ko \(\vec{b} + (-\vec{b}) = \vec{0}\).
Kōrero:
Tuatahi, ka tautuhia e mātou te whārite whakamuri o \(\vec{b}\):
\[
-\vec{b} = (-7, 5, 0)
\]
Muri iho, ka manatokohia e mātou ko te tāpiritanga o te whārite \(\vec{b}\) me tōna whārite whakamuri ka hua ake te whārite kore:
\[
\vec{b} + (-\vec{b}) = (7, -5, 0) + (-7, 5, 0)
\]
Ka tāpirihia e mātou ngā wāhanga o te vector:
\[
(7 – 7, -5 + 5, 0 + 0) = (0, 0, 0)
\]
Nō reira, \(\vec{b} + (-\vec{b}) = \vec{0}\), kua whakamātauhia ko te hua o te tāpiritanga o te whārite \(\vec{b}\) me tōna whārite whakamuri ko te whārite kore.
Tauira Pātai 3
I runga i ngā whārite \(\vec{u} = (2, -1)\) me \(\vec{v} = (-2, 1)\). Ko \(\vec{u}\) te whārite whakamuri o \(\vec{v}\) ?
Kōrero:
Hei whakatau mēnā he whārite whakamuri a \(\vec{u}\) me \(\vec{v}\), me tirotiro tātou mēnā ko \(\vec{v} = -\vec{u}\).
Tātaihia \(-\vec{u}\):
\[
-\vec{u} = (-2, 1)
\]
Ko te tikanga \(-\vec{u} = \vec{v}\), ko te tikanga ko te whārite \(\vec{u}\) ko te whārite whakamuri o te whārite \(\vec{v}\).
Tauira Pātai 4
Mena e mōhiotia ana ko te rahi o te whārite \(\vec{w}\) he 5, ā, he ahunga kei te ritenga kē atu i te whārite \(\vec{p} = (4, 3)\), whakatauhia \(\vec{w}\) i roto i te āhua wāhanga.
Kōrero:
Tuatahi, ka kitea e tātou te rahi o te whārite \(\vec{p}\):
\[
|\vec{p}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]
Nā te mea he rite te rahi o \(\vec{w}\) ki te \(\vec{p}\) engari he rerekē te ahunga, kāti:
\[
\vec{w} = -\vec{p} = (-4, -3)
\]
Nō reira, ko te whārite \(\vec{w}\) i roto i te āhua wāhanga ko \(\vec{w} = (-4, -3)\).
Tauira Pātai 5
Hoatu te pūwāhi A(2, 3) me te pūwāhi B(4, 7). Tāutuhia te ine tūnga mai i te pūwāhi A ki te pūwāhi B me te ine e anga ke ana ki taua ine.
Kōrero:
Te taunga tūnga mai i te pūwāhi A ki te pūwāhi B:
\[
\vec{AB} = (B_x – A_x, B_y – A_y) = (4 – 2, 7 – 3) = (2, 4)
\]
Ko te whārite whakamuri o \(\vec{AB}\):
\[
-\vec{AB} = (-2, -4)
\]
Nō reira, ko te whārite e whakahuri ana i te whārite tūnga \(\vec{AB} = (2, 4)\) ko \(-\vec{AB} = (-2, -4)\).
Tauira Pātai 6
Ki te hoatu he whārite \(\vec{m} = (x, y)\) ā, ko te whārite whakamuri o \(\vec{m}\) ko \( (-5, 12)\). Whakatauhia ngā uara o x me y.
Kōrero:
Ko te whārite whakamuri o \(\vec{m}\) ko \( (-x, -y) \), ā, e ai ki te raruraru, \((-x, -y) = (-5, 12)\).
Mā te whakataurite i ngā wāhanga whārite, ka whiwhi tātou:
\[
-x = -5 \e tohu ana x = 5
\]
\[
-y = 12 \e tohu ana y = -12
\]
Nō reira, ko te uara o \(x\) he 5, ā, ko te uara o \(y\) he -12.
Whakamutunga
Ko ngā whārite whakamuri he whārite he rite te rahi engari he rerekē te ahunga. Mā te mārama ki te ariā o ngā whārite whakamuri, ka taea e tātou te whakaoti i ngā raruraru e pā ana ki ngā whārite, pērā i te whakatau i te kino o tētahi whārite, te manatoko i te tāpiri o ngā whārite ki te kore, me ētahi atu. Ko te matapaki i ngā tauira raruraru i runga ake nei e tumanakohia ana ka whakapiki ake i tō tātou mōhiotanga me te māramatanga ki te mahi me ngā whārite whakamuri.
