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

Transcript

Misc 3 A girl walks 4 km towards west, then she walks 3 km in a direction 30ยฐ east of north and stops. Determine the girlโ€™s displacement from her initial point of departure.Let initial point of departure be O(0, 0) & OA = 4km, AB = 3km โˆ  XAB = 30ยฐ Here, Displacement will be vector (๐‘‚๐ต) โƒ— So, we need to find coordinates of B Now, โˆ  BAO = 90ยฐ โ€“ โˆ  XAB = 90ยฐ โ€“ 30ยฐ = 60ยฐ Let us draw BD โŠฅ OA In right triangle ADB sin 60ยฐ = ๐ต๐ท/๐ด๐ต โˆš3/2 = ๐ต๐ท/3 (3โˆš3)/2 = BD BD = (๐Ÿ‘โˆš๐Ÿ‘)/๐Ÿ In right triangle ADB cos 60ยฐ = ๐ด๐ท/๐ด๐ต 1/2 = ๐ด๐ท/3 3/2 = AD AD = ๐Ÿ‘/๐Ÿ Now, OD = OA โ€“ AD OD = 4 โ€“ ๐Ÿ‘/๐Ÿ = (๐Ÿ’(๐Ÿ) โˆ’ ๐Ÿ‘)/๐Ÿ = ๐Ÿ“/๐Ÿ So, x-coordinate of point B = โˆ’OD = (โˆ’๐Ÿ“)/๐Ÿ Thus, coordinates of point B = ((โˆ’๐Ÿ“)/๐Ÿ, (๐Ÿ‘โˆš๐Ÿ‘)/๐Ÿ) So, Displacement = (โˆ’๐Ÿ“)/๐Ÿ ๐’Š ฬ‚ + (๐Ÿ‘โˆš๐Ÿ‘)/๐Ÿ ๐’‹ ฬ‚ y-coordinate of point B = BD = (๐Ÿ‘โˆš๐Ÿ‘)/๐Ÿ Given, ๐‘Ž โƒ— = ๐‘ โƒ— + ๐‘ โƒ— Let ๐‘ โƒ— = 1๐‘– ฬ‚ + 2๐‘— ฬ‚ + 3๐‘˜ ฬ‚ & ๐‘ โƒ— = 2๐‘– ฬ‚ โˆ’ 1๐‘— ฬ‚ โˆ’ 2๐‘˜ ฬ‚ ๐‘Ž โƒ— = (๐‘ โƒ— + ๐‘ โƒ—) = (1 + 2) ๐‘– ฬ‚ + (2 โˆ’ 1) ๐‘— ฬ‚ + (3 โˆ’ 2) ๐‘˜ ฬ‚ = 3๐‘– ฬ‚ + 1๐‘— ฬ‚ + 1๐‘˜ ฬ‚ So, ๐‘Ž โƒ— = 3๐‘– ฬ‚ + 1๐‘— ฬ‚ + 1๐‘˜ ฬ‚ Magnitude of ๐‘Ž โƒ— = โˆš(32+1^2+1^2 ) |๐‘Ž โƒ— | = โˆš(9+1+1) = โˆš11 Given, ๐‘Ž โƒ— = ๐‘ โƒ— + ๐‘ โƒ— Let ๐‘ โƒ— = 1๐‘– ฬ‚ + 2๐‘— ฬ‚ + 3๐‘˜ ฬ‚ & ๐‘ โƒ— = 2๐‘– ฬ‚ โˆ’ 1๐‘— ฬ‚ โˆ’ 2๐‘˜ ฬ‚ ๐‘Ž โƒ— = (๐‘ โƒ— + ๐‘ โƒ—) = (1 + 2) ๐‘– ฬ‚ + (2 โˆ’ 1) ๐‘— ฬ‚ + (3 โˆ’ 2) ๐‘˜ ฬ‚ = 3๐‘– ฬ‚ + 1๐‘— ฬ‚ + 1๐‘˜ ฬ‚ So, ๐‘Ž โƒ— = 3๐‘– ฬ‚ + 1๐‘— ฬ‚ + 1๐‘˜ ฬ‚ Magnitude of ๐‘Ž โƒ— = โˆš(32+1^2+1^2 ) |๐‘Ž โƒ— | = โˆš(9+1+1) = โˆš11 Magnitude of ๐‘ โƒ— = โˆš(12+22+32) |๐‘ โƒ— | = โˆš(1+4+9) = โˆš14 Magnitude of ๐‘ โƒ— = โˆš(22+(โˆ’1)2+(โˆ’2)2) |๐‘ โƒ— | = โˆš(4+1+4) = โˆš9 = 3 Now, |๐‘ โƒ— | + |๐‘ โƒ— | = โˆš14 + 3 โ‰  โˆš11 โ‰  |๐‘Ž โƒ— | So, |๐‘Ž โƒ— |โ‰  |๐‘ โƒ— | + |๐‘ โƒ— | Hence, false.

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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.