He ariā nui ngā whārite i roto i te ahupūngao, e whakamahia ana hei tohu i ngā rahinga me te rahi me te ahunga. I roto i te ahupūngao, he maha ngā wā ka whakamahia ngā whārite hei whakaahua i ngā āhuatanga rerekē pēnei i te kaha, te tere, te whakaterenga, me ētahi atu. Ka matapakihia e tēnei tuhinga ētahi tauira o ngā raruraru whārite ahupūngao, me ō rātou otinga me ngā whakamārama.
1. Te Tāpiri me te Tangohanga o te Wētera
Tauira Pātai 1:
E rua ngā whārite \(\mathbf{A}\) me \(\mathbf{B}\) e hoatu ana penei:
\[
\mathbf{A} = 3\mathbf{i} + 4\mathbf{j}
\]
\[
\mathbf{B} = -2\mathbf{i} + 5\mathbf{j}
\]
Tātaihia:
1. \(\mathbf{A} + \mathbf{B}\)
2. \(\mathbf{A} – \mathbf{B}\)
Otinga:
Hei tāpiri i ngā wetere e rua, ka tāpirihia motuhaketia ō rāua wāhanga.
1. \(\mathbf{A} + \mathbf{B}\):
\[
\mathbf{A} + \mathbf{B} = (3\mathbf{i} + 4\mathbf{j}) + (-2\mathbf{i} + 5\mathbf{j})
\]
\[
= (3 – 2)\mathbf{i} + (4 + 5)\mathbf{j}
\]
\[
= 1\mathbf{i} + 9\mathbf{j}
\]
\[
\mathbf{A} + \mathbf{B} = \mathbf{i} + 9\mathbf{j}
\]
2. \(\mathbf{A} – \mathbf{B}\):
\[
\mathbf{A} – \mathbf{B} = (3\mathbf{i} + 4\mathbf{j}) – (-2\mathbf{i} + 5\mathbf{j})
\]
\[
= (3 – (-2))\mathbf{i} + (4 – 5)\mathbf{j}
\]
\[
= (3 + 2)\mathbf{i} + (-1)\mathbf{j}
\]
\[
= 5\mathbf{i} – \mathbf{j}
\]
Nā, ko te hua:
\[
\mathbf{A} – \mathbf{B} = 5\mathbf{i} – \mathbf{j}
\]
2. Whakarea Tauine (Whakaputa Ira)
Tauira Pātai 2:
E rua ngā pūwero \(\mathbf{C}\) me \(\mathbf{D}\) e hoatu ana penei:
\[
\mathbf{C} = 6\mathbf{i} + 2\mathbf{j}
\]
\[
\mathbf{D} = 3\mathbf{i} + 4\mathbf{j}
\]
Tātaihia te hua tauine (hua ira) o \(\mathbf{C}\) me \(\mathbf{D}\).
Otinga:
Ko te hua tauine o ngā whārite e rua \(\mathbf{C}\) me \(\mathbf{D}\) ko:
\[
\mathbf{C} \cdot \mathbf{D} = (6\mathbf{i} + 2\mathbf{j}) \cdot (3\mathbf{i} + 4\mathbf{j})
\]
\[
= 6 \cdot 3 + 2 \cdot 4
\]
\[
= 18 + 8
\]
\[
= 26
\]
Nō reira, ko te hua o te hua tauine o \(\mathbf{C}\) me \(\mathbf{D}\) ko 26.
3. Hua Whakawhiti
Tauira Pātai 3:
E rua ngā pūwero \(\mathbf{E}\) me \(\mathbf{F}\) e hoatu ana penei:
\[
\mathbf{E} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}
\]
\[
\mathbf{F} = 4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}
\]
Tātaihia te hua whakawhiti o \(\mathbf{E}\) me \(\mathbf{F}\).
Otinga:
Ka taea te tatau i te hua whakawhiti o ngā whārite e rua \(\mathbf{E}\) me \(\mathbf{F}\) mā te whakamahi i te whakatau whārite:
\[
\mathbf{E} \times \mathbf{F} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 me te 2 me te 3
4 & 5 & 6
\end{vmatrix}
\]
Tātaihia te whakatau o te matihiko:
\[
\mathbf{E} \times \mathbf{F} = \mathbf{i} (2 \cdot 6 – 3 \cdot 5) – \mathbf{j} (1 \cdot 6 – 3 \cdot 4) + \mathbf{k} (1 \cdot 5 – 2 \cdot 4)
\]
\[
= \mathbf{i} (12 – 15) – \mathbf{j} (6 – 12) + \mathbf{k} (5 – 8)
\]
\[
= \mathbf{i} (-3) – \mathbf{j} (-6) + \mathbf{k} (-3)
\]
\[
= -3\mathbf{i} + 6\mathbf{j} – 3\mathbf{k}
\]
Nō reira, ko te hua o te hua whakawhiti o \(\mathbf{E}\) me \(\mathbf{F}\) ko:
\[
\mathbf{E} \times \mathbf{F} = -3\mathbf{i} + 6\mathbf{j} – 3\mathbf{k}
\]
4. Te Rahi o te Wētera
Tauira Pātai 4:
I runga i te whārite \(\mathbf{G} = 3\mathbf{i} – 4\mathbf{j}\). Tātaihia te rahi (roa) o te whārite \(\mathbf{G}\).
Otinga:
Ka taea te tatau i te rahi o te whārite \(\mathbf{G}\) mā te whakamahi i te tātai:
\[
|\mathbf{G}| = \sqrt{(3)^2 + (-4)^2}
\]
\[
= \sqrt{9 + 16}
\]
\[
= \sqrt{25}
\]
\[
= 5
\]
Nō reira, ko te rahi o te whārite \(\mathbf{G}\) he 5.
5. Taumira Wetere
Tauira Pātai 5:
Ko te rahi o te whārite \(\mathbf{H}\) he 10 waeine, ā, he koki 30° te hanga ki te tuaka-x. Tāutuhia ngā wāhanga o te whārite \(\mathbf{H}\) i runga i ngā tuaka-x me ngā tuaka-y.
Otinga:
Ka taea te tatau i ngā wāhanga o te whārite \(\mathbf{H}\) i runga i ngā tuaka x (\(\mathbf{H}_x\)) me y (\(\mathbf{H}_y\)) mā te whakamahi i te ine whārite:
\[
\mathbf{H}_x = |\mathbf{H}| \cos(\theta)
\]
\[
\mathbf{H}_y = |\mathbf{H}| \hara(\theta)
\]
Me \(|\mathbf{H}| = 10\) me \(\theta = 30°\):
\[
\mathbf{H}_x = 10 \cos(30°)
\]
\[
\mathbf{H}_y = 10 \sin(30°)
\]
Ko ngā uara o \(\cos(30°) = \frac{\sqrt{3}}{2}\) me \(\sin(30°) = \frac{1}{2}\):
\[
\mathbf{H}_x = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}
\]
\[
\mathbf{H}_y = 10 \cdot \frac{1}{2} = 5
\]
Nō reira, ko ngā wāhanga o te whārite \(\mathbf{H}\) ko:
\[
\mathbf{H}_x = 5\sqrt{3}
\]
\[
\mathbf{H}_y = 5
\]
Whakamutunga
I roto i tēnei tuhinga, kua matapakihia e mātou ētahi tauira raruraru e pā ana ki ngā whārite i roto i te ahupūngao, mai i te tāpiri me te tango whārite, te whakarea tauine me te whakawhiti, ki te rahi me te taumira whārite. He mea nui te mārama ki te ariā me te mahi a ngā whārite i roto i te ahupūngao nā te mea he maha ngā āhuatanga taiao ka taea te whakamārama mā te whakamahi i ngā whārite. Ko te tumanako, mā ēnei tauira raruraru koe e āwhina ki te mārama ake i te ariā o ngā whārite.
