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  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise

Transcript

Ex 10.3, 15 (Introduction) If the vertices A, B, C of a triangle ABC are (1,2,3), (โ€“1, 0, 0), (0, 1, 2) respectively, then find โˆ ABC. [โˆ ABC is the angle between the vectors (๐ต๐ด) โƒ— and (๐ต๐ถ) โƒ—]. Consider a triangle ABC as shown โˆ  ABC is not the angle between vectors (๐ด๐ต) โƒ— and (๐ต๐ถ) โƒ— But the angle between vectors (๐ต๐ด) โƒ— and (๐ต๐ถ) โƒ— โˆด โˆ  ABC = Angle between vectors (๐ต๐ด) โƒ— and (๐ต๐ถ) โƒ— Ex 10.3, 15 If the vertices A, B, C of a triangle ABC are (1,2,3), (โ€“1, 0, 0), (0, 1, 2) respectively, then find โˆ ABC. [โˆ ABC is the angle between the vectors (๐ต๐ด) ฬ… and (๐ต๐ถ) ฬ…]. A (1, 2, 3) B (โˆ’1, 0, 0) C (0, 1, 2) โˆ ABC = Angle b/w (๐ต๐ด) โƒ— and (๐ต๐ถ) โƒ— We use formula ๐‘Ž โƒ—. ๐‘ โƒ— = |๐‘Ž โƒ— | |๐‘ โƒ— | cos ฮธ , ฮธ is the angle b/w ๐‘Ž โƒ— & ๐‘ โƒ— We find (๐ต๐ด) โƒ— and (๐ต๐ถ) โƒ— (๐ต๐ด) โƒ— = (1 โˆ’ (-1)) ๐‘– ฬ‚ + (2 โ€“ 0) ๐‘— ฬ‚ + (3 โ€“ 0) ๐‘˜ ฬ‚ = 2๐‘– ฬ‚ + 2๐‘— ฬ‚ + 3๐‘˜ ฬ‚ (๐ต๐ถ) โƒ— = (0 โˆ’ (โˆ’1)) ๐‘– ฬ‚ + (1 โˆ’ 0) ๐‘— ฬ‚ + (2 โˆ’ 0) ๐‘˜ ฬ‚ = 1๐‘– ฬ‚ + 1๐‘— ฬ‚ + 2๐‘˜ ฬ‚ Now, (๐‘ฉ๐‘จ) โƒ— . (๐‘ฉ๐‘ช) โƒ— = ("2" ๐‘– ฬ‚" + " 2๐‘— ฬ‚" + " 3๐‘˜ ฬ‚) . ("1" ๐‘– ฬ‚" + " 1๐‘— ฬ‚" + " 2๐‘˜ ฬ‚) = 2.1+2.1+3.2 = 2+2+6 = 10 Magnitude of (๐ต๐ถ) โƒ— = โˆš(12+12+22) |(๐‘ฉ๐‘ช) โƒ— | = โˆš(1+1+4) = โˆš๐Ÿ” Also, (๐ต๐ด) โƒ— . (๐ต๐ถ) โƒ— = |(๐ต๐ด) โƒ— | . |(๐ต๐ถ) โƒ— | cos ฮธ 10 = โˆš17 ร— โˆš6 ร— cos ฮธ โˆš17 ร— โˆš6 ร— cos ฮธ = 10 cos ฮธ = 10/(โˆš17 ร—โˆš6) cos ฮธ = 10/โˆš102 ฮธ = cosโˆ’1 (10/โˆš102) Thus โˆ ABC = cosโˆ’1 (๐Ÿ๐ŸŽ/โˆš๐Ÿ๐ŸŽ๐Ÿ).

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.