Ngā Tauira Pātai me te Kōrero mō te Tangohanga Wekita
Pendahuluan
I roto i te pāngarau me te ahupūngao, he ariā matua ngā whārite e whakamahia ana hei whakamārama i ngā āhuatanga taiao me ngā āhuatanga hangarau maha. Ko te whārite he rahinga he nui, he ahunga hoki. Ko ētahi tauira nui o ngā whārite ko te nekehanga, te tere, te whakaterenga, me te kaha. I roto i tēnei tuhinga, ka matapakihia e tātou te tango whārite, ahakoa he maha ngā wā ka whakanuia tēnei kaupapa i roto i te horopaki o te whakakotahitanga whārite.
He mahi taketake te tango whārite e tino hira ana i roto i te tātari whārite. Hei ruku hohonu ake ki tēnei ariā, me arotake ētahi tauira raruraru me ngā kōrero e pā ana ki te tango whārite.
Te Tangohanga Wetereo
Ko te tango whārite {\displaystyle \mathbf{A} – \mathbf{B}} e tautuhia ana ko te mahi tāpiri i te whārite {\displaystyle \mathbf{A}} me te whārite {\displaystyle -\mathbf{B}}, ko {\displaystyle -\mathbf{B}} he whārite he rite te rahi ki {\displaystyle \mathbf{B}} engari he ritenga kē. Mā te pāngarau, ka taea te tuhi pēnei:
{\displaystyle \mathbf{A} – \mathbf{B} = \mathbf{A} + (-\mathbf{B})}
Ngā Pātai Tauira me te Kōrero
Pātai 1: Te Tango i ngā Wētera Ahu-Rua
Mehemea e rua ngā wekere i roto i ngā taunga Cartesian:
{\displaystyle \mathbf{A} = (4, 3)} me {\displaystyle \mathbf{B} = (1, 2)}. Tātaihia {\displaystyle \mathbf{A} – \mathbf{B}}.
Kōrero:
Ko te taahiraa tuatahi ko te kimi i te whārite kino o {\displaystyle \mathbf{B}}, arā:
{\displaystyle -\mathbf{B} = (-1, -2)}
Muri iho, tāpirihia te whārite {\displaystyle \mathbf{A}} me te {\displaystyle -\mathbf{B}}:
{\displaystyle \mathbf{A} – \mathbf{B} = (4, 3) + (-1, -2)}
Tāpirihia ia wāhanga x me y mā te whakamahi i te tāpiritanga whārite:
{\displaystyle \mathbf{A} – \mathbf{B} = (4 + (-1), 3 + (-2))}
{\displaystyle \mathbf{A} – \mathbf{B} = (3, 1)}
Nō reira, ko te hua o te tango i ngā whārite {\displaystyle \mathbf{A} – \mathbf{B}} ko te whārite (3, 1).
Pātai 2: Te Tango i ngā Wētera Ahu-Toru
E rua ngā whārite i roto i ngā taunga toru-ahu e hoatu ana:
{\displaystyle \mathbf{P} = (2, -4, 6)} me {\displaystyle \mathbf{Q} = (-3, 5, 7)}. Tātaihia {\displaystyle \mathbf{P} – \mathbf{Q}}.
Kōrero:
Ko te taahiraa tuatahi ko te kimi i te whārite kino o {\displaystyle \mathbf{Q}}:
{\displaystyle -\mathbf{Q} = (3, -5, -7)}
Muri iho, tāpirihia te whārite {\displaystyle \mathbf{P}} me te {\displaystyle -\mathbf{Q}}:
{\displaystyle \mathbf{P} – \mathbf{Q} = (2, -4, 6) + (3, -5, -7)}
Tāpirihia ia wāhanga x, y, me z mā te whakamahi i te tāpiritanga whārite:
{\displaystyle \mathbf{P} – \mathbf{Q} = (2 + 3, -4 + (-5), 6 + (-7))}
{\displaystyle \mathbf{P} – \mathbf{Q} = (5, -9, -1)}
Nō reira, ko te hua o te tango i ngā whārite {\displaystyle \mathbf{P} – \mathbf{Q}} ko te whārite (5, -9, -1).
Pātai 3: Te Tangohanga Wekita i te Papa Matatini
Me kī e rua ngā whārite e tohuhia ana e ngā tau matatini:
{\displaystyle \mathbf{M} = 3 + 4i} me {\displaystyle \mathbf{N} = 1 + 2i}. Tātaihia {\displaystyle \mathbf{M} – \mathbf{N}}.
Kōrero:
Ko te taahiraa tuatahi ko te kimi i te whārite kino o {\displaystyle \mathbf{N}}:
{\displaystyle -\mathbf{N} = -1 – 2i}
Muri iho, tāpirihia te whārite {\displaystyle \mathbf{M}} me te {\displaystyle -\mathbf{N}}:
{\displaystyle \mathbf{M} – \mathbf{N} = (3 + 4i) + (-1 – 2i)}
Tāpirihia ia wāhanga tūturu me ia wāhanga pohewa mā te whakamahi i te tāpiritanga whārite:
{\displaystyle \mathbf{M} – \mathbf{N} = (3 + (-1)) + (4i + (-2i))}
{\displaystyle \mathbf{M} – \mathbf{N} = 2 + 2i}
Nō reira, ko te hua o te tango i ngā whārite {\displaystyle \mathbf{M} – \mathbf{N}} ko te tau matatini 2 + 2i.
Pātai 4: Te Tangohanga Wetereo i roto i te Pūnaha Taunga Pōro
Me kī e rua ngā whārite i roto i ngā taunga porowhita:
Ko te rahi o {\displaystyle \mathbf{U}} he 5, ā, ko te koki he 30°,
ā, ko te rahi o {\displaystyle \mathbf{V}} he 3, ā, ko te koki he 150°.
Tātaihia {\displaystyle \mathbf{U} – \mathbf{V}}.
Kōrero:
Ko te taahiraa tuatahi ko te huri i ngā whārite {\displaystyle \mathbf{U}} me {\displaystyle \mathbf{V}} ki ngā taunga Cartesian.
Mo {\displaystyle \mathbf{U}}:
{\displaystyle U_x = 5 \cos(30^\circ) = 5 \left(\frac{\sqrt{3}}{2}\right) = 5 \cdot 0.866 = 4.33}
{\displaystyle U_y = 5 \sin(30^\circ) = 5 \left(\frac{1}{2}\right) = 5 \cdot 0.5 = 2.5}
Nō reira, ko te {\displaystyle \mathbf{U}} i roto i te Cartesian ko (4.33, 2.5).
Mō {\displaystyle \mathbf{V}}:
{\displaystyle V_x = 3 \cos(150^\circ) = 3 \left(\frac{-\sqrt{3}}{2}\right) = 3 \cdot (-0.866) = -2.598}
{\displaystyle V_y = 3 \sin(150^\circ) = 3 \left(\frac{1}{2}\right) = 3 \cdot 0.5 = 1.5}
Nō reira, ko te {\displaystyle \mathbf{V}} i roto i te Cartesian ko (-2.598, 1.5).
Ko te taahiraa e whai ake nei, tatauhia te tangohanga whārite i roto i te Cartesian:
{\displaystyle \mathbf{U} – \mathbf{V} = (4.33, 2.5) – (-2.598, 1.5)}
Ko te tikanga mā te tāpiri i te kino o te vector:
{\displaystyle \mathbf{U} – \mathbf{V} = (4.33 + 2.598, 2.5 – 1.5)}
{\displaystyle \mathbf{U} – \mathbf{V} = (6.928, 1)}
Nō reira, ko te hua o te tango i te whārite {\displaystyle \mathbf{U} – \mathbf{V}} i roto i ngā taunga Cartesian ko (6.928, 1).
Whakamutunga
He mahi pāngarau nui te tango i ngā whārite i roto i ngā mara maha e whakamahi ana i te tātari whārite. Ahakoa i roto i ngā pūnaha taunga rua-ahu, toru-ahu, uaua, porowhita rānei, he rite tonu te kaupapa matua: te tāpiri i tētahi whārite ki te taha kino o tētahi atu. E whakaatu ana ngā tauira i runga ake nei i ngā huarahi maha hei whakamahi i tēnei mahi i roto i ngā horopaki rerekē, hei āwhina i a tātou ki te mārama ake i te ariā me te mahi.
