Ngā Tauira Pātai me te Kōrero mō ngā Wētera Ahu-Toru i roto i te Pūnaha Taunga Cartesian
He ariā nui ngā whārite toru-ahu i roto i te pāngarau me te ahupūngao, e whakamahia ana hei tohu i ngā mea, i ngā āhuatanga rānei i roto i te wāhi toru-ahu. I roto i te pūnaha taunga Cartesian, e toru ngā wāhanga o ēnei whārite, e tohuhia ana ko \( (x, y, z) \). Ka matapakihia e tēnei tuhinga ētahi tauira raruraru me ngā otinga e pā ana ki ngā whārite toru-ahu i roto i te pūnaha taunga Cartesian.
Te Mārama ki ngā Wētera Ahu-Toru
Ka taea te whakaatu i tētahi whārite i roto i te wāhi toru-ahu penei \(\mathbf{A} = (A_x, A_y, A_z)\), ina:
– Ko \(A_x\) te wāhanga o te ira i te tuaka-x.
– Ko \(A_y\) te wāhanga whārite i te tuaka-y.
– Ko \(A_z\) te wāhanga whārite i te tuaka-z.
Ngā Pātai Tauira me te Kōrero
Pātai 1: Te Mahi Tāpiri Wētera
E rua ngā whārite e hoatu ana, \(\mathbf{A} = (2, -3, 4)\) me \(\mathbf{B} = (-1, 5, 2)\). Tātaihia te tapeke o ēnei whārite e rua.
Kōrero:
Ko te tāpiri i ngā whārite e rua \(\mathbf{A}\) me \(\mathbf{B}\) ka mahia mā te tāpiri i ō rāua wāhanga e rite ana. Nō reira, kei a tātou:
\[
\mathbf{C} = \mathbf{A} + \mathbf{B} = (A_x + B_x, A_y + B_y, A_z + B_z)
\]
Whakakapia ngā uara whārite kua hoatu:
\[
\mathbf{C} = (2 + (-1), -3 + 5, 4 + 2) = (1, 2, 6)
\]
Nō reira, ko te hua o te tāpiri i ngā whārite \(\mathbf{A}\) me \(\mathbf{B}\) ko \(\mathbf{C} = (1, 2, 6)\).
Pātai 2: Te Mahi Tangohanga Wekete
E rua ngā whārite e hoatu ana, \(\mathbf{A} = (4, 1, -2)\) me \(\mathbf{B} = (5, -3, 6)\). Tātaihia te tangohanga o ēnei whārite e rua, arā, \(\mathbf{A} – \mathbf{B}\).
Kōrero:
Ko te tango i ngā whārite e rua \(\mathbf{A}\) me \(\mathbf{B}\) ka mahia mā te tango i ō rāua wāhanga e rite ana. Nō reira, kei a tātou:
\[
\mathbf{D} = \mathbf{A} – \mathbf{B} = (A_x – B_x, A_y – B_y, A_z – B_z)
\]
Whakakapia ngā uara whārite kua hoatu:
\[
\mathbf{D} = (4 – 5, 1 – (-3), -2 – 6) = (-1, 4, -8)
\]
Nō reira, ko te hua o te tango i ngā whārite \(\mathbf{A}\) me \(\mathbf{B}\) ko \(\mathbf{D} = (-1, 4, -8)\).
Pātai 3: Te Mahi Whakarea Tauine
Hoatu he whārite \(\mathbf{A} = (3, -2, 7)\) me te tauine \(k = 4\). Tātaihia te hua tauine o ēnei whārite.
Kōrero:
Ko te whakarea i tētahi tauine \(k\) ki tētahi whārite \(\mathbf{A}\) ka mahia mā te whakarea i ia wāhanga o te whārite ki taua taurite. Nō reira, kei a tātou:
\[
\mathbf{E} = k \cdot \mathbf{A} = k \cdot (A_x, A_y, A_z) = (k \cdot A_x, k \cdot A_y, k \cdot A_z)
\]
Whakakapia ngā uara kua hoatu:
\[
\mathbf{E} = 4 \cdot (3, -2, 7) = (4 \cdot 3, 4 \cdot -2, 4 \cdot 7) = (12, -8, 28)
\]
Nō reira, ko te hua o te whakarea i te tauine \(k\) ki te whārite \(\mathbf{A}\) ko \(\mathbf{E} = (12, -8, 28)\).
Pātai 4: Te Roa o te Wetere
Tātaihia te roa (rahi) o te whārite \(\mathbf{A} = (1, 2, 2)\).
Kōrero:
Ka taea te tatau i te roa, te rahi rānei o tētahi whārite \(\mathbf{A} = (A_x, A_y, A_z)\) mā te whakamahi i te tātai:
\[
|\mathbf{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}
\]
Whakakapia ngā uara kua hoatu:
\[
|\mathbf{A}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3
\]
Nō reira, ko te roa o te whārite \(\mathbf{A}\) he 3.
Pātai 5: Hua Ira
E rua ngā whārite e hoatu ana, \(\mathbf{A} = (1, 0, -1)\) me \(\mathbf{B} = (2, 3, 4)\). Tātaihia te hua ira o ēnei whārite e rua.
Kōrero:
Ka oti te hua ira o ngā whārite e rua \(\mathbf{A} = (A_x, A_y, A_z)\) me \(\mathbf{B} = (B_x, B_y, B_z)\) mā te whakarea i ngā wāhanga e rite ana, kātahi ka tāpirihia. Nō reira, kei a tātou:
\[
\mathbf{A} \cdot \mathbf{B} = A_x \cdot B_x + A_y \cdot B_y + A_z \cdot B_z
\]
Whakakapia ngā uara kua hoatu:
\[
\mathbf{A} \cdot \mathbf{B} = (1 \cdot 2) + (0 \cdot 3) + (-1 \cdot 4) = 2 + 0 – 4 = -2
\]
Nō reira, ko te hua ira o ngā whārite \(\mathbf{A}\) me \(\mathbf{B}\) he -2.
Pātai 6: Hua Whakawhiti
E rua ngā whārite e hoatu ana, \(\mathbf{A} = (1, 2, 3)\) me \(\mathbf{B} = (4, 5, 6)\). Tātaihia te hua whakawhiti o ēnei whārite e rua.
Kōrero:
Ka oti te hua whakawhiti o ngā whārite e rua \(\mathbf{A} = (A_x, A_y, A_z)\) me \(\mathbf{B} = (B_x, B_y, B_z)\) mā te whakamahi i te tātai e whai ake nei:
\[
\mathbf{A} \times \mathbf{B} = \left( (A_y \cdot B_z – A_z \cdot B_y), (A_z \cdot B_x – A_x \cdot B_z), (A_x \cdot B_y – A_y \cdot B_x) \right)
\]
Whakakapia ngā uara kua hoatu:
\[
\mathbf{A} \times \mathbf{B} = \left( (2 \cdot 6 – 3 \cdot 5), (3 \cdot 4 – 1 \cdot 6), (1 \cdot 5 – 2 \cdot 4) \right) = (12 – 15, 12 – 6, 5 – 8) = (-3, 6, -3)
\]
Nō reira, ko te hua whakawhiti o ngā whārite \(\mathbf{A}\) me \(\mathbf{B}\) ko \(\mathbf{A} \times \mathbf{B} = (-3, 6, -3)\).
Whakamutunga
He taputapu nui ngā whārite toru-ahu i roto i te pūnaha taunga Cartesian i roto i ngā momo mara o te pūtaiao me te hangarau. Mā roto i ngā tauira me ngā kōrero i runga ake nei, kua kite tātou me pēhea te mahi i ngā mahi taketake i runga i ngā whārite, pērā i te tāpiri, te tango, te whakarea tauine, me ngā hua ira me te whakawhiti. He mea tino nui te māramatanga pakari ki ēnei ariā, ehara i te mea i roto i te pāngarau anake, engari i roto hoki i ngā tono mahi i roto i te ahupūngao, te hangarau, me te pūtaiao rorohiko.
