Misali Tambaya Ta Tattaunawa Kan Ƙarin Vectors Biyu Ta Amfani da Hanyar Parallelogram
Ƙarin Vector muhimmin ra'ayi ne a fannin kimiyyar lissafi da lissafi, wanda galibi ana amfani da shi don bayyana abubuwan da suka faru na halitta da matsalolin rayuwar yau da kullun. Akwai hanyoyi da yawa don ƙara vector guda biyu, ɗaya daga cikinsu shine hanyar parallelogram. Wannan hanyar ba wai kawai tana da sauƙin fahimta ba ce, har ma tana ba da kyakkyawan hangen nesa na yadda vector guda biyu ke haɗuwa don samar da vector mai sakamako. A cikin wannan labarin, za mu duba misalai da yawa na ƙarin vector ta amfani da hanyar parallelogram, tare da mafita.
Menene Vektor?
Kafin mu shiga cikin matsalolin misalan, muna buƙatar fahimtar ma'anar vector ta asali. Vector adadi ne wanda ke da girma (tsawo) da alkibla. Misalan vector na gargajiya sun haɗa da gudu, hanzari, ƙarfi, da ƙaura. Ana iya wakiltar vector a matsayin abubuwan da ke cikinsa (i, j, k) a cikin daidaitattun Cartesian ko kuma a matsayin tsayinsa da alkiblarsa (kusurwar).
Hanyar Parallelogram
Hanyar parallelogram hanya ɗaya ce ta ƙara vectors guda biyu. A cikin wannan hanyar, muna wakiltar vectors guda biyu a matsayin ɓangarorin parallelogram guda biyu. Vector mai sakamakon shine diagonal na parallelogram wanda ya fara daga wurin farawa na vectors guda biyu. A lissafi, idan muna da vectors guda biyu \(\vec{A}\) da \(\vec{B}\), sakamakon shine \( \vec{R} = \vec{A} + \vec{B} \).
Hanyar mataki-mataki don amfani da hanyar parallelogram ita ce kamar haka:
1. Zana vector \(\vec{A}\) daga wurin farawa.
2. Daga ƙarshen vector \(\vec{A}\), zana vector \(\vec{B}\).
3. Zana layi mai layi ɗaya da vector \(\vec{B}\) daga wurin farawa \(\vec{A}\).
4. Zana layi mai layi ɗaya da vector \(\vec{A}\) daga ƙarshen vector \(\vec{B}\).
5. Zana diagonal daga wurin farawa zuwa kusurwar da ke gaba da juna don samun vector mai sakamakon \(\vec{R}\).
Tambayoyi da Tattaunawa Samfura
Tambaya ta 1
A ce muna da vectors guda biyu \(\vec{A}\) da \(\vec{B}\):
– \(\vec{A}\) yana da tsayi (girma) na raka'a 5 da kuma alkiblar 0° (ko kuma tare da axis mai kyau na x),
– \(\vec{B}\) yana da tsawon raka'a 3 da kuma alkiblar 90° (ko kuma a kan axis mai kyau na y).
Menene amfanin ƙara waɗannan vectors guda biyu ta amfani da hanyar parallelogram?
Tattaunawa:
1. Zana vector \(\vec{A}\) tare da axis mai kyau na x tare da tsawon raka'a 5.
2. Daga ƙarshen vector \(\vec{A}\), zana vector \(\vec{B}\) tare da axis mai kyau na y tare da tsawon raka'a 3.
3. Daga wurin farawa \(\vec{A}\), zana layi a layi ɗaya da \(\vec{B}\).
4. Daga ƙarshen \(\vec{B}\), zana layi a layi ɗaya da \(\vec{A}\).
5. Sakamakon shine parallelogram mai diagonal wanda shine vector mai sakamakon \(\vec{R}\).
Tunda \(\vec{A}\) da \(\vec{B}\) suna tsaye a tsaye, za mu iya amfani da ka'idar Pythagorean don ƙididdige tsawon vector ɗin da ya biyo baya:
\[ R = \sqrt{A^2 + B^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \kimanin 5.83 \]
Ana iya ƙididdige alkiblar vector ɗin da aka samu ta amfani da trigonometry. Idan \(\theta\) shine kusurwar da ke tsakanin sakamakon da \(\vec{A}\):
\[ \tan(\theta) = \frac{B}{A} = \frac{3}{5} \]
haka:
\[ \theta = \tan^{-1}\left(\frac{3}{5}\right) \kimanin 30.96^\circ \]
Saboda haka, vector mai sakamakon \(\vec{R}\) yana da girman kusan raka'a 5.83 kuma alkiblar kusan 30.96° daga \(\vec{A}\).
Tambaya ta 2
An bayar da vector guda biyu \(\vec{C}\) da \(\vec{D}\) kamar haka:
– \(\vec{C}\) mai tsawon raka'a 4 da kuma alkiblar 45°.
– \(\vec{D}\) mai tsawon raka'a 6 da kuma alkiblar 120°.
Kayyade vector ɗin da aka samu daga ƙarin vector guda biyu.
Tattaunawa:
Don ƙara vector guda biyu waɗanda ba su daidaita da juna ko kuma a siffofi daban-daban ba, zaku iya amfani da abubuwan Cartesian.
1. Raba \(\vec{C}\) da \(\vec{D}\) zuwa sassan x da y.
Domin \(\vec{C}\):
\[ C_x = C \cos(45^\circ) = 4 \cos(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2} \kimanin 2.83 \]
\[ C_y = C \sin(45^\circ) = 4 \sin(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2} \kimanin 2.83 \]
Domin \(\vec{D}\):
\[ D_x = D \cos(120^\circ) = 6 \cos(120^\circ) = 6 \cdot (-\frac{1}{2}) = -3 \]
\[ D_y = D \sin(120^\circ) = 6 \sin(120^\circ) = 6 \cdot \frac{\sqrt{3}}{2} = 3\sqrt{3} \kimanin 5.20 \]
2. Ƙara sassan x da y na vectors guda biyu:
\[ R_x = C_x + D_x = 2.83 + (-3) = -0.17 \]
\[ R_y = C_y + D_y = 2.83 + 5.20 = 8.03 \]
3. Lissafa girma da alkiblar vector ɗin da aka samu \(\vec{R}\):
\[ R = \sqrt{R_x^2 + R_y^2} = \sqrt{(-0.17)^2 + 8.03^2} = \sqrt{0.03 + 64.48} = \sqrt{64.51} \kimanin 8.03 \]
\[ \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{8.03}{-0.17}\right) \approx \tan^{-1}(-47.24) \]
Tunda sakamakon ba shi da kyau, muna ƙara 180° don samun kusurwar a cikin tsarin kwata-kwata daidai:
\[ \theta \approx \tan^{-1}(47.24) + 180^\cir \approx 271.93^\cir \]
Don haka, vector mai sakamakon \(\vec{R}\) yana da girman kusan raka'a 8.03 da kuma alkiblar kusan 271.93°, ko kuma za mu iya cewa kimanin 91.93° daga axis ɗin x mara kyau a cikin kwata na huɗu.
Penutup
Hanyar parallelogram hanya ce mai inganci da gani don ƙara vectors guda biyu. Duk da cewa wannan hanyar na iya zama kamar mai sauƙi ga vectors masu sauƙi, yana da mahimmanci a fahimci cewa ga vectors masu rikitarwa, sau da yawa muna buƙatar amfani da abubuwan Cartesian da dabarun algebra masu ci gaba don samun sakamako masu kyau. Da fatan, misalan da ke sama suna ba da cikakken hoto na yadda za a iya amfani da wannan hanyar a yanayi daban-daban.
