УРАВНЕНИЯ ПЛОСКОСТИ И ПРЯМОЙ В ПРОСТРАНСТВЕ

Аннотация

This work studies the analytical representation of plane and straight line equations in space, their types, geometric interpretation, and mutual localization cases. In a three-dimensional coordinate system in space, general, point-vector, and normal equations of the plane are introduced. Also, parametric and direction vector equations of a straight line are covered. During the work, the relationship between a plane and a straight line, the conditions for their intersection, and the cases of parallelism and perpendicularity are analyzed. Also, attention is paid to important geometric concepts such as the distance from a point to a plane or straight line, angle, and projection. This topic occupies an important place in the analytical geometry section of mathematics and is widely used in the fields of engineering, physics, and computer graphics. As a result of the work, students will acquire the skills to represent and analyze the main geometric objects in space in an analytical form. Keywords: Plane in space, straight line, coordinate system, equation of the plane, normal vector, parametric equation, vector equation, direction vector, intersection of planes, mutual location of a line and a plane, parallel and perpendicular lines, distance from a point to a plane, angle in space, analytical geometry, projection, spatial analysis. Language: Uzbek 152 Universaljurnal.uz Universal International Scientific Journal 2025, 2(2) Citation: Azimov , R. (2025). EQUATIONS OF A PLANE AND A STRAIGHT LINE IN SPACE. Universal International Scientific Journal, 2(2), 151–159. Retrieved from https://universaljurnal.uz/index.php/jurnal/article/view/1493 Doi: https://doi.org/10.5281/zenodo.15228628 Google scholar: https://scholar.google.co.in/scholar?hl=ru&as_sdt=0%2C5&q=YOSHLARNING+MA%27NAVIYAT+DUNYOQARASHINI+RIVOJLANTIRISHDA+UZLUKSIZ+TA%27LIM+JARAYONLARINING+O%27RNI+%28RIVOJLANGAN+DAVLATLAR+TAJRIBASI+MISOLIDA%29&btnG= Kirish. Analitik geometriya Ax+By+Cz+D=0 fazodagi geometrik obyektlarning (1.1) algebraik ifodalanishini o‘rganadi va ularning o‘zaro bog‘liqliklarini aniqlashda va 𝑎⃗ = {𝑎 ,𝑎 ,𝑎 }, berilgan (1.1) 1 2 3 muhim vosita hisoblanadi. Ayniqsa, uch tekislikka parallel bo‘lsin (1-rasm), o‘lchamli Evklid fazosida tekislik va to‘g‘ri chiziq kabi asosiy geometrik elementlarning tenglamalarini aniqlash va ularni tahlil qilish zamonaviy matematikaning dolzarb yo‘nalishlaridan biridir. Fazoda tekislik va to‘g‘ri chiziqning analitik ifodasi, ularning 1-rasm. parametrik, vektorli hamda umumiy Bu xolda (1.1) tekislikdan shunday tenglama ko‘rinishlari, bu obyektlar ikkita 𝑀 va 𝑀 nuqtalar topadaliki bular orasidagi geometrik munosabatlar 1 2 (masalan, kesishish, parallel va uchun, ⃗𝑀⃗⃗⃗⃗⃗⃗𝑀⃗⃗⃗⃗⃗= 𝑎⃗ bo‘ladi. Shartga ko‘ra 1 2 perpendikulyarlik) turli sohalarda — 𝑀 (𝑥 𝑢 𝑧 ) va 𝑀 (𝑥 𝑢 𝑧 ) nuqtalar 1 1 1 1 2 2 2 2 xususan, muhandislik, fizik berilgan tekislikda yotganligi uchun modellashtirish, kompyuter grafikasi va 𝐴𝑥 + 𝐵𝑦 + 𝐶𝑧 + 𝐷 = 0 va 1 1 1 fazoviy tahlil ishlanmalarida keng 𝐴𝑥 + 𝐵𝑦 + 𝐶𝑧 + 𝐷 = 0 2 2 2 qo‘llaniladi. Mazkur maqolada fazoda shartlar bajariladi, bundan esa tekislik va to‘g‘ri chiziq tenglamalarining quyidagicha tenglikka ega bo‘lamiz: asosiy turlari, ularning o‘zaro joylashuv holatlari, shuningdek, nuqta, chiziq va 𝐴(𝑥 − 𝑥 ) + 𝐵(𝑦 − 𝑦 ) + 𝐶(𝑧 − 𝑧 ) 2 1 2 1 2 1 tekislik orasidagi masofalar, burchaklar va = 0 proyeksiyalarni aniqlash usullari tizimli ravishda bayon etiladi. Tadqiqot natijalari Ikkinchi tomondan 𝑀⃗⃗⃗⃗⃗⃗⃗𝑀⃗⃗⃗⃗⃗ = {𝑥 − 1 2 2 fazoviy geometriya masalalarining nazariy 𝑥 ,𝑦 − 𝑦 ,𝑧 − 𝑧 } = 𝑎⃗ ekanligidan 1 2 1 2 1 asoslarini mustahkamlash va ularni quyidagi tenglikka ega bo‘lamiz: amaliyotga tatbiq etishda foydali bo‘lishi A𝑎 + B𝑎 + C𝑎 = 0 1 2 3 mumkin. (1.2) Bu xosil bo‘lgan (1.2) shart 𝑎⃗ Tekislikning umumiy tenglamasi. vektorni berilgan tekislikka parallellik Vektorning tekislikka parallelik sharti. shartini ifodalaydi. Tekislik o‘zining umumiy tenglamasi bilan berilgan bo‘lsin. 153 Universaljurnal.uz Universal International Scientific Journal 2025, 2(2) Tekislikning umumiy tenglamasini fazodagi vaziyatini o‘rganamiz. a) Agar A≠ 0, B≠ 0, C≠ 0, D= 0 bo‘lsa, u xolda (1.1) tenglamamiz quyidagi ko‘rinishda bo‘ladi: Ax+By+Cz+D=0 Bu xolda tekisligimiz koordinata boshidan o‘tuvchi tekislik bo‘ladi, chunki O(0,0,0) nuqtaning koordinatalari tekislik 4 – rasm. tenglamasini qanoatlantiradi (2 -rasm). r) Faraz qilaylik A≠ 0, B≠ 0, C= 0, D≠ 0 bo‘lsin, u xolda tenglamamiz quyidagi ko‘rinishda bo‘ladi: Ax+D=0. Bu xolda xosil bo‘lgan tekislik tenglamasi bilan 𝑗⃗ = {0,1,0} va 𝑘 ⃗⃗ = {0,0,1} vektorlar uchun (1.2) shart bajariladi. Tekisligimiz koordinata boshidan o‘tmaganligi uchun (D≠ 0 ) , u (0 y z) tekisligiga parallel tekislik 2-rasm. b o ‘ l a d i ( 5- rasm). 3-rasm. b) Faraz qilaylik A≠ 0, B≠ 0, D= 0, C≠ 0 bo‘lsin, u xolda tenglamamiz quyidagi ko‘rinishda bo‘ladi: Ax+By+D=0. Bu xolda 𝑘 ⃗⃗ = {0,0,1} vektor bilan (1.1) tekislik uchun (1.2) shart bajariladi, bundan esa tekislik oz o‘qiga parallel ekanligi kelib chiqadi. (