Three Dimensional Geometry
Madhya Pradesh Board · Class 12 · Mathematics
Flashcards for Three Dimensional Geometry — Madhya Pradesh Board Class 12 Mathematics. Quick Q&A cards covering key concepts, definitions, and formulas.
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Get startedएक ऐसी रेखा के दिशा कोसाइन ज्ञात कीजिए जिसके दिशा अनुपात 2, -1, -2 हैं।
Answer
चरण 1: दिशा अनुपात से दिशा कोसाइन के लिए सूत्र का उपयोग कीजिए. l = ± a / √(a² + b² + c²), m = ± b / √(a² + b² + c²), n = ± c / √(a² + b² + c²) चरण 2: a = 2, b = -1, c = -2 रखने पर प्राप्त कीजिए. √(…
Find the direction cosines of a line whose direction ratios are 2, -1, -2.
Answer
Step 1: Use the formula for direction cosines from direction ratios. l = ± a / √(a² + b² + c²), m = ± b / √(a² + b² + c²), n = ± c / √(a² + b² + c²) Step 2: Substitute a = 2, b = -1, c = -2. √(2² +…
A line makes angles 90°, 60°, and 30° with the positive x-, y-, and z-axes. Find its direction cosines.
Answer
Step 1: Direction cosines are the cosines of the direction angles. Step 2: Compute each value. l = cos 90° = 0 m = cos 60° = 1/2 n = cos 30° = √3/2 Answer: 0, 1/2, √3/2…
एक रेखा धनात्मक x-, y-, और z-अक्षों के साथ 90°, 60°, और 30° के कोण बनाती है। इसके दिशा कोसाइन ज्ञात कीजिए।
Answer
चरण 1: दिशा कोसाइन दिशा कोणों के कोसाइन होते हैं. चरण 2: प्रत्येक मान निकालिए. l = cos 90° = 0 m = cos 60° = 1/2 n = cos 30° = √3/2 उत्तर: 0, 1/2, √3/2…
रेखा की दिशा उलटने पर उसके दिशा कोसाइन के चिह्न क्यों बदल जाते हैं?
Answer
एक निर्देशित रेखा के अक्षों के साथ कोण α, β, γ होते हैं। दिशा उलटने पर प्रत्येक कोण उसके पूरक में बदल जाता है: π - α, π - β, π - γ. तब: cos(π - α) = -cos α cos(π - β) = -cos β cos(π - γ) = -cos γ इस…
Why does reversing the direction of a line change the signs of its direction cosines?
Answer
A directed line has angles α, β, γ with the axes. Reversing the direction changes each angle to its supplement: π - α, π - β, π - γ. Then: cos(π - α) = -cos α cos(π - β) = -cos β cos(π - γ) = -cos γ …
What are direction ratios, and why are they not unique?
Answer
Any three numbers proportional to the direction cosines of a line are direction ratios. If a, b, c are direction ratios, then ka, kb, kc for k ≠ 0 are also direction ratios. Example: If 2, -1, -2 ar…
दिशा अनुपात क्या होते हैं, और वे अद्वितीय क्यों नहीं होते?
Answer
कोई भी तीन संख्याएँ जो किसी रेखा के दिशा कोसाइन के समानुपाती हों, दिशा अनुपात कहलाती हैं. यदि a, b, c दिशा अनुपात हैं, तो k ≠ 0 के लिए ka, kb, kc भी दिशा अनुपात होंगे. उदाहरण: यदि 2, -1, -2 दिशा अनु…
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