Tauira o ngā Raru e Matapaki ana i ngā Wētera Tūnga
He ariā taketake ngā whārite i roto i te pāngarau me te ahupūngao, e tohu ana i ngā rahinga me te ahunga me te rahi. I roto i ngā tono maha, ka whakamahia ngā whārite hei whakaahua i te tūranga, te tere, te kaha, me te maha atu o ngā tawhā. I roto i ngā momo whārite, he mahi nui tā ngā whārite tūranga ki te mahere i te taunga o tētahi pūwāhi i te wāhi.
Te Whakamāramatanga o te Wētera Tūnga
Ko te ira tūnga he ira e whakaahua ana i te taunga o tētahi pūwāhi e pā ana ki te pūtake i roto i tētahi pūnaha taunga. I te nuinga o te wā, ka tuhia he ira tūnga ki te āhua taunga Cartesian penei:
\[ \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \]
I konei, ko \(\mathbf{r}\) te ira tūnga, ko \(x\), \(y\), me \(z\) ōna wāhanga i te taha o ngā tuaka \(x\), \(y\), me \(z\), ia, ko \(\mathbf{i}\), \(\mathbf{j}\), me \(\mathbf{k}\) ia he ira wae e whakarara ana ki ngā tuaka taunga. I roto i te wāhi rua-ahu, kāore te wāhanga \(z\) i te nuinga o te wā e noho ana, nō reira ka noho te ira tūnga:
\[ \mathbf{r} = x\mathbf{i} + y\mathbf{j} \]
Ngā Taupānga Wāhitau Tūnga
Hei tauira, i roto i te ahupūngao, he mea nui te tūranga o ngā whārite i roto i te whakaahua i te nekehanga o ngā mea. Ka taea te whakaatu i te tūranga o te mea e pā ana ki te pūtake (te pūwāhi tohutoro) mā te whārite tūranga. Hei tāpiri, i roto i te miihini, he maha ngā wā ka whakamahia ngā whārite tūranga hei tatau i ngā kaha me ngā wā.
Ngā Tauira Pātai me te Kōrero mō ngā Wētera Tūnga
Pātai 1
Me kī e rua ngā pūwāhi i te wāhi 3D, ko te pūwāhi A me ngā taunga \( (1, 2, 3) \) me te pūwāhi B me ngā taunga \( (4, 0, -2) \). Whakatauhia ngā whārite tūnga o ngā pūwāhi A me B. Hei tāpiri, tatauhia te whārite e hono ana i te pūwāhi A ki te pūwāhi B.
Kōrero:
Te tūnga o te pūwāhi A:
\[ \mathbf{r_A} = 1\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \]
Te tūnga o te pūwāhi B:
\[ \mathbf{r_B} = 4\mathbf{i} + 0\mathbf{j} – 2\mathbf{k} \]
Muri iho, hei kimi i te ira e hono ana i te pūwāhi A ki te pūwāhi B (e kiia nei ko \(\mathbf{AB}\)), me tango e tātou te ira tūnga o A mai i te ira tūnga o B:
\[ \mathbf{AB} = \mathbf{r_B} – \mathbf{r_A} \]
Nā, mā te whakakapi i ngā whārite tūnga e rua i runga ake nei:
\[ \mathbf{AB} = (4\mathbf{i} + 0\mathbf{j} – 2\mathbf{k}) – (1\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}) \]
\[ \mathbf{AB} = (4 – 1)\mathbf{i} + (0 – 2)\mathbf{j} + (-2 – 3)\mathbf{k} \]
\[ \mathbf{AB} = 3\mathbf{i} – 2\mathbf{j} – 5\mathbf{k} \]
Nō reira, ko te whārite e hono ana i te pūwāhi A ki te pūwāhi B ko \( 3\mathbf{i} – 2\mathbf{j} – 5\mathbf{k} \).
Pātai 2
Mena kei runga i te \((2, 3)\) i te papa 2D te pūwāhi P, kimihia te roa (noa) o te ira tūnga \(\mathbf{r_P}\).
Kōrero:
Te tūnga o te pūwāhi P:
\[ \mathbf{r_P} = 2\mathbf{i} + 3\mathbf{j} \]
Ka taea te tatau i te roa o te ira tūnga \(\mathbf{r_P}\) mā te whakamahi i te tātai ira noa (te roa rānei):
\[ \| \mathbf{r_P} \| = \sqrt{x^2 + y^2} \]
Whakakapia ngā uara o \(x\) me \(y\):
\[ \| \mathbf{r_P} \| = \sqrt{2^2 + 3^2} \]
\[ \| \mathbf{r_P} \| = \sqrt{4 + 9} \]
\[ \| \mathbf{r_P} \| = \sqrt{13} \]
Nō reira, ko te roa o te ira tūnga \(\mathbf{r_P}\) ko \(\sqrt{13}\).
Pātai 3
Me kī kei te \( (5, -4, 2) \) te pūwāhi Q. Kimihia te koki i waenganui i te ira tūnga \(\mathbf{r_Q}\) me te tuaka \(x\).
Kōrero:
Te tūnga o te pūwāhi Q:
\[ \mathbf{r_Q} = 5\mathbf{i} – 4\mathbf{j} + 2\mathbf{k} \]
Hei kimi i te koki i waenganui i te whārite \(\mathbf{r_Q}\) me te tuaka \(x\), ka taea e tātou te whakamahi i te ariā o te hua ira. Tuatahi, ka whakatauhia e tātou te hua ira i waenganui i \(\mathbf{r_Q}\) me \(\mathbf{i}\):
\[ \mathbf{r_Q} \cdot \mathbf{i} = 5\mathbf{i} \cdot \mathbf{i} + (-4\mathbf{j} \cdot \mathbf{i}) + 2\mathbf{k} \cdot \mathbf{i} \]
Nā te mea ko \(\mathbf{i} \cdot \mathbf{i} = 1\), \(\mathbf{j} \cdot \mathbf{i} = 0\), me \(\mathbf{k} \cdot \mathbf{i} = 0\), kāti:
\[ \mathbf{r_Q} \cdot \mathbf{i} = 5 \]
Te paerewa o \(\mathbf{r_Q}\):
\[ \| \mathbf{r_Q} \| = \sqrt{5^2 + (-4)^2 + 2^2} \]
\[ \| \mathbf{r_Q} \| = \sqrt{25 + 16 + 4} \]
\[ \| \mathbf{r_Q} \| = \sqrt{45} \]
\[ \| \mathbf{r_Q} \| = 3\sqrt{5} \]
Ko te paerewa o \(\mathbf{i}\) he 1, nā te mea he whārite waeine a \(\mathbf{i}\).
Mā te whakamahi i te tātai hua ira hei kimi i te koki \(\theta\):
\[ \mathbf{r_Q} \cdot \mathbf{i} = \| \mathbf{r_Q} \| \| \mathbf{i} \| \cos\theta\]
\[ 5 = 3\sqrt{5} \cos\theta \]
\[ \cos\theta = \frac{5}{3\sqrt{5}} \]
\[ \cos\theta = \frac{5}{3\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} \]
\[ \cos\theta = \frac{5\sqrt{5}}{15} \]
\[ \cos\theta = \frac{\sqrt{5}}{3} \]
Nō reira, ko te koki \(\theta\) i waenganui i te ira tūnga \(\mathbf{r_Q}\) me te tuaka \(x\) ko:
\[ \theta = \cos^{-1} \left(\frac{\sqrt{5}}{3}\right) \]
Whakamutunga
He mea nui te mahi a ngā whārite tūnga i roto i te pūtaiao me te hangarau, inā koa i te mahere i te tūnga o ngā mea i roto i te wāhi taunga. E whakaatu ana ngā tauira i runga ake nei me pēhea te tatau i ngā whārite tūnga, ō rātou roa, me ngā koki i waenganui i a rātou me ngā tuaka taunga. He mea tino nui te mārama ki ēnei ariā taketake hei whakaoti rapanga maha e pā ana ki te wāhi me ngā taunga i roto i te pāngarau me te ahupūngao.
