Tha vectaran nam bun-bheachd cudromach ann am fiosaig, air an cleachdadh gus àireamhan a riochdachadh le meud agus stiùireadh. Ann am fiosaig, bidh vectaran gu tric air an cleachdadh gus diofar nithean leithid feachd, astar, luathachadh, agus barrachd a mhìneachadh. Bruidhnidh an artaigil seo air grunn eisimpleirean de dhuilgheadasan vectar fiosaig, còmhla ris na fuasglaidhean agus na mìneachaidhean aca.
1. Cur-ris agus Toirt-às Vectar
Eisimpleir Ceist 1:
Tha dà vectar \(\mathbf{A}\) agus \(\mathbf{B}\) air an toirt seachad mar a leanas:
\[
\mathbf{A} = 3\mathbf{i} + 4\mathbf{j}
\]
\[
\mathbf{B} = -2\mathbf{i} + 5\mathbf{j}
\]
Obraich a-mach:
1. \(\mathbf{A} + \mathbf{B}\)
2. \(\mathbf{A} – \mathbf{B}\)
Fuasgladh:
Gus dà vectar a chur ri chèile, bidh sinn a’ cur an co-phàirtean air leth.
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. \(\mathbf{A} – \mathbf{B}\):
\[
\mathbf{A} - \mathbf{B} = (3\mathbf{i} + 4\mathbf{j}) - (-2\mathbf{i} + 5\mathbf{j})
\]
\[
= (3 – (-2))\mathbf{i} + (4 – 5)\mathbf{j}
\]
\[
= (3 + 2)\mathbf{i} + (-1)\mathbf{j}
\]
\[
= 5\mathbf{i} – \mathbf{j}
\]
Mar sin, is e an toradh:
\[
\mathbf{A} – \mathbf{B} = 5\mathbf{i} – \mathbf{j}
\]
2. Iomadachadh Sgalar (Toradh Dot)
Eisimpleir Ceist 2:
Tha dà vectar \(\mathbf{C}\) agus \(\mathbf{D}\) air an toirt seachad mar a leanas:
\[
\mathbf{C} = 6\mathbf{i} + 2\mathbf{j}
\]
\[
\mathbf{D} = 3\mathbf{i} + 4\mathbf{j}
\]
Obraich a-mach an toradh sgalar (toradh dot) de \(\mathbf{C}\) agus \(\mathbf{D}\).
Fuasgladh:
Is e toradh sgalar dà vectar \(\mathbf{C}\) agus \(\mathbf{D}\) a th’ ann:
\[
\mathbf{C} \cdot \mathbf{D} = (6\mathbf{i} + 2\mathbf{j}) \cdot (3\mathbf{i} + 4\mathbf{j})
\]
\[
= 6 ⋅ 3 + 2 ⋅ 4
\]
\[
= 18 + 8
\]
\[
= 26
\]
Mar sin, is e 26 toradh toradh sgalar \(\mathbf{C}\) agus \(\mathbf{D}\).
3. Tar-thoradh
Eisimpleir Ceist 3:
Tha dà vectar \(\mathbf{E}\) agus \(\mathbf{F}\) air an toirt seachad mar a leanas:
\[
\mathbf{E} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}
\]
\[
\mathbf{F} = 4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}
\]
Obraich a-mach an toradh-croise de \(\mathbf{E}\) agus \(\mathbf{F}\).
Fuasgladh:
Faodar toradh-croise dà vectar \(\mathbf{E}\) agus \(\mathbf{F}\) obrachadh a-mach a’ cleachdadh an determinant maitrís:
\[
\mathbf{E} \times \mathbf{F} = \toiseach{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 & 2 & 3 \\
4 & 5 & 6
\end{vmatrix}
\]
Obraich a-mach determinant a’ mhaitrice:
\[
\mathbf{E} \times \mathbf{F} = \mathbf{i} (2 \cdot 6 – 3 \cdot 5) – \mathbf{j} (1 \cdot 6 – 3 \cdot 4) + \mathbf{k} (1 \cdot 5 – 2) \cdot 5 – 2 \cdot
\]
\[
= \mathbf{i} (12 – 15) – \mathbf{j} (6 – 12) + \mathbf{k} (5 – 8)
\]
\[
= \mathbf{i} (-3) – \mathbf{j} (-6) + \mathbf{k} (-3)
\]
\[
= -3\mathbf{i} + 6\mathbf{j} - 3\mathbf{k}
\]
Mar sin, is e toradh an tar-thoraidh de \(\mathbf{E}\) agus \(\mathbf{F}\):
\[
\mathbf{E} \times \mathbf{F} = -3\mathbf{i} + 6\mathbf{j} - 3\mathbf{k}
\]
4. Meudachd Vectar
Eisimpleir Ceist 4:
Ma tha an vectar \(\mathbf{G} = 3\mathbf{i} – 4\mathbf{j}\) air a thoirt seachad, obraich a-mach meud (fad) a’ vectar \(\mathbf{G}\).
Fuasgladh:
Faodar meud a’ vectar \(\mathbf{G}\) obrachadh a-mach leis an fhoirmle:
\[
|\mathbf{G}| = \sqrt{(3)^2 + (-4)^2}
\]
\[
= \sqrt{9 + 16}
\]
\[
= 25
\]
\[
= 5
\]
Mar sin, tha meud a’ vectar \(\mathbf{G}\) co-ionann ri 5.
5. Fuasgladh Vectar
Eisimpleir Ceist 5:
Tha meud 10 aonadan aig a’ vectar H agus tha e a’ cruthachadh ceàrn 30° leis an ais-x. Obraich a-mach co-phàirtean a’ vectar H air na h-aisealan-x agus y.
Fuasgladh:
Faodar co-phàirtean a’ vectar H air na h-aisealan x (H_x) agus y (H_y) obrachadh a-mach le bhith a’ cleachdadh triantanachd:
\[
\mathbf{H}_x = |\mathbf{H}| \ cos(\theta)
\]
\[
\mathbf{H}_y = |\mathbf{H}| \sin(\theta)
\]
Le \(|\mathbf{H}| = 10\) agus \(\theta = 30°\):
\[
H_x = 10 cos(30°)
\]
\[
H_y = 10 sin(30°)
\]
Na luachan aig \(cos(30°) = \frac{\sqrt{3}}{2}\) agus \(sin(30°) = \frac{1}{2}\):
\[
\mathbf{H}_x = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}
\]
\[
H_y = 10 \cdot \frac{1}{2} = 5
\]
Mar sin, is iad seo co-phàirtean a’ vectar \(\mathbf{H}\):
\[
H_x = 5 ³
\]
\[
H_y = 5
\]
Co-dhùnadh
San artaigil seo, tha sinn air grunn eisimpleirean de dhuilgheadasan a dheasbad a tha a’ toirt a-steach vectaran ann am fiosaig, bho chur-ris is toirt-air-falbh vectaran, iomadachadh sgalar is crois, gu meud is rùn vectaran. Tha tuigse air bun-bheachd agus obrachadh vectaran deatamach ann am fiosaig oir faodar mòran de thachartasan nàdarra a mhìneachadh le bhith a’ cleachdadh vectaran. Tha sinn an dòchas gun cuidich na eisimpleirean de dhuilgheadasan seo thu gus bun-bheachd vectaran a thuigsinn nas doimhne.
