Ama-vector angumqondo obalulekile ku-physics, asetshenziselwa ukumela ubuningi ngobukhulu kanye nesiqondiso. Ku-physics, ama-vector avame ukusetshenziswa ukuchaza izenzakalo ezahlukahlukene njengamandla, ijubane, ukusheshisa, nokuningi. Lesi sihloko sizoxoxa ngezibonelo eziningana zezinkinga zama-vector ze-physics, kanye nezixazululo zazo kanye nezincazelo.
1. Ukwengezwa Nokususwa Kwevektha
Isibonelo Sombuzo 1:
Amavekhtha amabili \(\mathbf{A}\) kanye \(\mathbf{B}\) anikezwa kanje:
\[
\mathbf{A} = 3\mathbf{i} + 4\mathbf{j}
\]
\[
\mathbf{B} = -2\mathbf{i} + 5\mathbf{j}
\]
Bala:
1. \(\mathbf{A} + \mathbf{B}\)
2. \(\mathbf{A} – \mathbf{B}\)
Isixazululo:
Ukuze sengeze amavekhtha amabili, sengeza izingxenye zawo ngokwehlukana.
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. \(\mathbf{A} – \mathbf{B}\):
\[
\mathbf{A} – \mathbf{B} = (3\mathbf{i} + 4\mathbf{j}) – (-2\mathbf{i} + 5\mathbf{j})
\]
\[
= (3 – (-2))\mathbf{i} + (4 – 5)\mathbf{j}
\]
\[
= (3 + 2)\mathbf{i} + (-1)\mathbf{j}
\]
\[
= 5\mathbf{i} – \mathbf{j}
\]
Ngakho-ke, umphumela uba:
\[
\mathbf{A} – \mathbf{B} = 5\mathbf{i} – \mathbf{j}
\]
2. Ukuphindaphinda kwe-Scalar (Umkhiqizo we-Dot)
Isibonelo Sombuzo 2:
Amavekhtha amabili \(\mathbf{C}\) kanye \(\mathbf{D}\) anikezwa kanje:
\[
\mathbf{C} = 6\mathbf{i} + 2\mathbf{j}
\]
\[
\mathbf{D} = 3\mathbf{i} + 4\mathbf{j}
\]
Bala umkhiqizo we-scalar (umkhiqizo we-dot) we-\(\mathbf{C}\) kanye ne-\(\mathbf{D}\).
Isixazululo:
Umkhiqizo we-scalar wamavektha amabili \(\mathbf{C}\) kanye \(\mathbf{D}\) uthi:
\[
\mathbf{C} \cdot \mathbf{D} = (6\mathbf{i} + 2\mathbf{j}) \cdot (3\mathbf{i} + 4\mathbf{j})
\]
\[
= 6 \cdot 3 + 2 \cdot 4
\]
\[
= 18 + 8
\]
\[
= 26
\]
Ngakho-ke, umphumela womkhiqizo we-scalar we-\(\mathbf{C}\) kanye ne-\(\mathbf{D}\) u-26.
3. Umkhiqizo Ohlanganisiwe
Isibonelo Sombuzo 3:
Amavekhtha amabili \(\mathbf{E}\) kanye \(\mathbf{F}\) anikezwa kanje:
\[
\mathbf{E} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}
\]
\[
\mathbf{F} = 4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}
\]
Bala umkhiqizo ohlanganisiwe we-\(\mathbf{E}\) kanye ne-\(\mathbf{F}\).
Isixazululo:
Umkhiqizo ohlanganisiwe wamavekhtha amabili \(\mathbf{E}\) kanye \(\mathbf{F}\) ungabalwa kusetshenziswa isichazi-matrix:
\[
\mathbf{E} \izikhathi \mathbf{F} = \qala{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 kanye no-2 kanye no-3 \\
4 & 5 & 6
\end{vmatrix}
\]
Bala isichazi se-matrix:
\[
\mathbf{E} \izikhathi \mathbf{F} = \mathbf{i} (2 \cdot 6 – 3 \cdot 5) – \mathbf{j} (1 \cdot 6 – 3 \cdot 4) + \mathbf{k} (1 \cdot 5 – 4 \cdot)
\]
\[
= \mathbf{i} (12 – 15) – \mathbf{j} (6 – 12) + \mathbf{k} (5 – 8)
\]
\[
= \mathbf{i} (-3) – \mathbf{j} (-6) + \mathbf{k} (-3)
\]
\[
= -3\mathbf{i} + 6\mathbf{j} – 3\mathbf{k}
\]
Ngakho-ke, umphumela womkhiqizo ohlanganisiwe we-\(\mathbf{E}\) kanye ne-\(\mathbf{F}\) uthi:
\[
\mathbf{E} \izikhathi \mathbf{F} = -3\mathbf{i} + 6\mathbf{j} – 3\mathbf{k}
\]
4. Ubukhulu beVektha
Isibonelo Sombuzo 4:
Uma ubheka i-vector \(\mathbf{G} = 3\mathbf{i} – 4\mathbf{j}\). Bala ubukhulu (ubude) be-vector \(\mathbf{G}\).
Isixazululo:
Ubukhulu bevektha \(\mathbf{G}\) bungabalwa kusetshenziswa ifomula:
\[
|\mathbf{G}| = \sqrt{(3)^2 + (-4)^2}
\]
\[
= \sqrt{9 + 16}
\]
\[
= \sqrt{25}
\]
\[
= 5
\]
Ngakho-ke, ubukhulu bevektha \(\mathbf{G}\) bungu-5.
5. Isixazululo seVektha
Isibonelo Sombuzo 5:
Ivektha \(\mathbf{H}\) inobukhulu obungamayunithi ayi-10 futhi yakha i-engeli engu-30° nge-x-axis. Thola izingxenye zevektha \(\mathbf{H}\) kuma-x- kanye nama-y-axis.
Isixazululo:
Izingxenye zevektha \(\mathbf{H}\) kuma-axes ka-x (\(\mathbf{H}_x\)) kanye no-y (\(\mathbf{H}_y\)) zingabalwa kusetshenziswa i-trigonometry:
\[
\mathbf{H}_x = |\mathbf{H}| \cos(\theta)
\]
\[
\mathbf{H}_y = |\mathbf{H}| \isono(\theta)
\]
Nge-\(|\mathbf{H}| = 10\) kanye ne-\(\theta = 30°\):
\[
\mathbf{H}_x = 10 \cos(30°)
\]
\[
\mathbf{H}_y = 10 \sin(30°)
\]
Amanani ka-\(\cos(30°) = \frac{\sqrt{3}}{2}\) kanye no-\(\sin(30°) = \frac{1}{2}\):
\[
\mathbf{H}_x = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}
\]
\[
\mathbf{H}_y = 10 \cdot \frac{1}{2} = 5
\]
Ngakho-ke, izingxenye zevektha \(\mathbf{H}\) yilezi:
\[
\mathbf{H}_x = 5\sqrt{3}
\]
\[
\mathbf{H}_y = 5
\]
Isiphetho
Kulesi sihloko, sixoxe ngezinkinga eziningana zezibonelo ezihilela ama-vector ku-physics, kusukela ekuhlanganiseni nasekususeni ama-vector, ekuphindaphindeni nasekuphindaphindeni okuphambene, kuya kubukhulu be-vector kanye nesisombululo. Ukuqonda umqondo kanye nokusebenza kwama-vector kubalulekile ku-physics ngoba izenzakalo eziningi zemvelo zingachazwa kusetshenziswa ama-vector. Ngethemba ukuthi lezi zinkinga zezibonelo zizokusiza uqonde umqondo wama-vector ngokujulile.
