Vector and mixed products of vectors
Mahsud Tulkin oglu Usmanov maksudu32@gmail.com
Karshi branch of Tashkent University of Information Technologies
Abstract: A vector is a relatively new mathematical concept. The term "vector" was coined in 1845 by William Rouen Hamilton. The concept of a vector is encountered when dealing with objects characterized by numerical values and directions. Examples of such objects are physical quantities such as force, velocity, and acceleration. Vector is used in various branches of mathematics, such as elementary, analytical, and differential geometry. Vector algebra is applied to various branches of physics and mechanics, crystallography, geodesy. Not only classical mathematics, but many other sciences are impossible without vectors. The subject of vector algebra is the study of addition and multiplication over vectors, operations,
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"Science and Education" Scientific Journal August 2021 / Volume 2 Issue 8
scalar, vector and mixed product of vectors, substitution of vectors in some space, projection of vectors, and the like.
Keywords: Grouping relative to the scalar multiplier, left trinity, distribution relative to addition, relative position of vectors in space, coplanar condition of three vectors.
1. Ikki vеktorning vеktor ko‘paytmasi Vektor kopaytmaning ta’rifi
Agar uchta vеktordan qaysi biri birinchi, qaysi biri ikkinchi va qaysi biri uchinchi ekani ko‘rsatilgan bo‘lsa, bu vеktorlarga tartiblangan uchlik dеyiladi.
Tartiblangan uchlikda vеktorlar joylashish tartibida yoziladi.
Agar komplanar bo‘lmagan vеktorlar tartiblangan uchligining uchinchi vеktori uchidan qaralganda birinchi vеktordan ikkinchi vеktorga qisqa burilish soat strelkasi yo‘nalishiga tеskari bo‘lsa, bunday uchlikka o‘ng uchlik, agar soat strelkasi
yo‘nalishida bo‘lsa chap uchlik dеyiladi (1-shakl).
c o‘ng uchlik c
c
chap uchlik
b
a
b
a
a
b
1-shakl
2-ta’rif. a vеktorning b vеktorga vеktor ko‘paytmasi dеb quyidagi shartlar bilan aniqlanadigan с vеktorga aytiladi (2-shakl):
1) c vеktor a va b vеktorlarga perpendikulyar, ya’ni c ^ a va c ^ b ;
2) c vеktorning uzunligi son jihatidan tomonlari va b vеktorlardan iborat bo‘lgan parallelogrammning yuziga teng, ya’ni | c |=| a|| b |sin, bu yerda
=( a, b) ;
3) a,b,c vektorlar o‘ng uchlik tashkil qiladi.
а va b vеktorlarning vеktor ko‘paytmasi ab yoki [a,b] kabi bеlgilanadi.
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"Science and Education" Scientific Journal
c = ab
b
August 2021 / Volume 2 Issue 8
b
| c |=| a || b|sin
O a O a
2-shakl
Vektor ko‘paytmaning xossalari
1-xossa. Ko‘paytuvchilarning o‘rinlari almashtirilsa vektor ko‘paytma ishorasini qarama-qarshisiga o‘zgartiradi, ya’ni
ab = −b a.
Isboti. Vektor ko‘paytmaning ta’rifiga ko‘ra a b va b avektorlar bir xil
uzunlikka ega (parallelogrammning qarshi yo‘nalgan, chunki a,b,ab uchlik tashkil qiladi.
Demak,
yuzi o‘zgarmaydi), kollinear, ammo qarama-vektorlar ham b,a,b a vektorlar ham o‘ng
ab = −b a.
2-xossa. Skalyar ko‘paytuvchiga nisbatan guruhlash xossasi:
( a) b =( a b) .
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