Isibonelo Sezinkinga Ezixoxa Ngezikhundla Ze-Vectors
Ama-vector angumqondo oyisisekelo kwizibalo kanye ne-physics, amelela ubuningi obuhambisana nesiqondiso kanye nobukhulu. Ezisetshenzisweni ezahlukene, ama-vector avame ukusetshenziswa ukuchaza isikhundla, ijubane, amandla, kanye neminye imingcele eminingi. Phakathi kwezinhlobo ezahlukene zama-vector, ama-vector esikhundla adlala indima ebalulekile ekudwebeni indawo yephuzu esikhaleni.
Incazelo yeVektha Yesikhundla
Ivektha yesikhundla iyivektha echaza indawo yephuzu maqondana nomsuka ohlelweni lwe-coordinate. Ngokuvamile, ivektha yesikhundla ibhalwa ngesimo se-Cartesian coordinate kanje:
\[ \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \]
Lapha, i-\(\mathbf{r}\) iyivektha yesikhundla, \(x\), \(y\), kanye ne-\(z\) yizingxenye zayo eceleni kwe-\(x\), \(y\), kanye ne-\(z\) axes, ngokulandelana, kuyilapho i-\(\mathbf{i}\), \(\mathbf{j}\), kanye ne-\(\mathbf{k}\) kuyivektha yeyunithi ehambisana ne-coordinate axes, ngokulandelana. Esikhaleni esinezinhlangothi ezimbili, ingxenye ye-\(z\) ngokuvamile ayikho, ngakho-ke ivektha yesikhundla iba:
\[ \mathbf{r} = x\mathbf{i} + y\mathbf{j} \]
Izicelo Zevektha Yesikhundla
Isibonelo, ku-physics, ama-position vectors adlala indima ebalulekile ekuchazeni ukunyakaza kwezinto. Indawo yento maqondana nomsuka (indawo yokubhekisela) ingamelwa yi-position vector. Ngaphezu kwalokho, kubunjiniyela bemishini, ukubalwa kwamandla nezikhathi kuvame ukuhilela ukusetshenziswa kwama-position vectors.
Imibuzo Yezibonelo kanye Nengxoxo Yama-Vector Ezikhundla
umbuzo 1
Ake sithi kunamaphuzu amabili esikhaleni se-3D, iphuzu A elinezixhumanisi \( (1, 2, 3) \) kanye nephuzu B elinezixhumanisi \( (4, 0, -2) \). Thola amavekhtha endawo yamaphuzu A no-B. Ngaphezu kwalokho, bala iphuzu lokuxhumanisa ivekhtha A nephuzu B.
Ingxoxo:
Ivektha yesikhundla sephuzu A:
\[ \mathbf{r_A} = 1\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \]
Ivektha yesikhundla sephuzu B:
\[ \mathbf{r_B} = 4\mathbf{i} + 0\mathbf{j} – 2\mathbf{k} \]
Okulandelayo, ukuthola i-vector connecting point A to point B (ebizwa ngokuthi \(\mathbf{AB}\)), sidinga ukususa i-position vector ka-A ku-position vector ka-B:
\[ \mathbf{AB} = \mathbf{r_B} – \mathbf{r_A} \]
Ngakho-ke, ukufaka esikhundleni ama-vector amabili esikhundla ngenhla:
\[ \mathbf{AB} = (4\mathbf{i} + 0\mathbf{j} – 2\mathbf{k}) – (1\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}) \]
\[ \mathbf{AB} = (4 – 1)\mathbf{i} + (0 – 2)\mathbf{j} + (-2 – 3)\mathbf{k} \]
\[ \mathbf{AB} = 3\mathbf{i} – 2\mathbf{j} – 5\mathbf{k} \]
Ngakho-ke, iphuzu lokuxhumanisa i-vector u-A no-B lingu-\( 3\mathbf{i} – 2\mathbf{j} – 5\mathbf{k} \).
umbuzo 2
Uma iphuzu u-P liku-\((2, 3)\) endizeni ye-2D, thola ubude (okujwayelekile) bevektha yesikhundla \(\mathbf{r_P}\).
Ingxoxo:
Ivektha yesikhundla sephuzu P:
\[ \mathbf{r_P} = 2\mathbf{i} + 3\mathbf{j} \]
Ubude bevektha yesikhundla \(\mathbf{r_P}\) bungabalwa kusetshenziswa ifomula ye-vektha evamile (noma ubude):
\[ \| \mathbf{r_P} \| = \sqrt{x^2 + y^2} \]
Faka amanani ka-\(x\) kanye no-\(y\):
\[ \| \mathbf{r_P} \| = \sqrt{2^2 + 3^2} \]
\[ \| \mathbf{r_P} \| = \sqrt{4 + 9} \]
\[ \| \mathbf{r_P} \| = \sqrt{13} \]
Ngakho-ke, ubude bevektha yesikhundla \(\mathbf{r_P}\) bungu \(\sqrt{13}\).
umbuzo 3
Ake sithi iphuzu u-Q lilele ku-\( (5, -4, 2) \). Thola i-engeli phakathi kwevektha yesikhundla \(\mathbf{r_Q}\) kanye ne-axis \(x\).
Ingxoxo:
Ivektha yesikhundla sephuzu Q:
\[ \mathbf{r_Q} = 5\mathbf{i} – 4\mathbf{j} + 2\mathbf{k} \]
Ukuze sithole i-engeli phakathi kwevektha \(\mathbf{r_Q}\) kanye ne-axis \(x\) axis, singasebenzisa umqondo womkhiqizo wechashazi. Okokuqala, sinquma umkhiqizo wechashazi phakathi \(\mathbf{r_Q}\) kanye \(\mathbf{i}\):
\[ \mathbf{r_Q} \cdot \mathbf{i} = 5\mathbf{i} \cdot \mathbf{i} + (-4\mathbf{j} \cdot \mathbf{i}) + 2\mathbf{k} \cdot \mathbf{i}]
Kusukela ku-\(\mathbf{i} \cdot \mathbf{i} = 1\), \(\mathbf{j} \cdot \mathbf{i} = 0\), kanye ne-\(\mathbf{k} \cdot \mathbf{i} = 0\), khona-ke:
\[ \mathbf{r_Q} \cdot \mathbf{i} = 5 \]
Isimiso se-\(\mathbf{r_Q}\):
\[ \| \mathbf{r_Q} \| = \sqrt{5^2 + (-4)^2 + 2^2} \]
\[ \| \mathbf{r_Q} \| = \sqrt{25 + 16 + 4} \]
\[ \| \mathbf{r_Q} \| = \sqrt{45} \]
\[ \| \mathbf{r_Q} \| = 3\sqrt{5} \]
Isilinganiso se-\(\mathbf{i}\) singu-1, ngoba i-\(\mathbf{i}\) iyi-vector yeyunithi.
Ukusebenzisa ifomula yomkhiqizo wamachashazi ukuthola i-engeli \(\theta\):
\[ \mathbf{r_Q} \cdot \mathbf{i} = \| \mathbf{r_Q} \| \| \mathbf{i} \| \cos\theta\]
\[ 5 = 3\sqrt{5} \cos\theta \]
\[ \cos\theta = \frac{5}{3\sqrt{5}} \]
\[ \cos\theta = \frac{5}{3\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} \]
\[ \cos\theta = \frac{5\sqrt{5}}{15} \]
\[ \cos\theta = \frac{\sqrt{5}}{3} \]
Ngakho-ke i-engeli \(\theta\) phakathi kwevektha yesikhundla \(\mathbf{r_Q}\) kanye ne-axis \(x\) yile:
\[ \theta = \cos^{-1} \left(\frac{\sqrt{5}}{3}\right) \]
Isiphetho
Ama-position vectors adlala indima ebalulekile kwisayensi nobunjiniyela, ikakhulukazi ekudwebeni indawo yezinto esikhaleni esihlanganisiwe. Izibonelo ezingenhla zibonisa indlela yokubala ama-position vectors, ubude bawo, kanye nama-engeli aphakathi kwawo kanye nama-coordinate axes. Ukuqonda le mibono eyisisekelo kubaluleke kakhulu ekuxazululeni izinkinga ezahlukahlukene ezihilela isikhala kanye nama-coordinate kwizibalo kanye ne-physics.
