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Misc 15 - Prove that (a + b) . (a - b) = |a|2 + |b|2 - Vector - Scalar product - Solving

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  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise
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Misc 15 Prove that (๐‘Žย  + ๐‘ย ) โ‹… (๐‘Žย +๐‘ย ) =|๐‘Žย โƒ—|2 + |๐‘ย โƒ—|2 , if and only if ๐‘Žย โƒ—, ๐‘ย โƒ— are perpendicular, given ๐‘Žย โƒ— โ‰  0ย โƒ—, ๐‘ย โƒ— โ‰  0ย โƒ— (๐‘Žย โƒ— + ๐‘ย โƒ—) โ‹… (๐‘Žย โƒ— + ๐‘ย โƒ—) = ๐‘Žย โƒ— . ๐‘Žย โƒ— + ๐‘Žย โƒ— . ๐‘ย โƒ— + ๐’ƒย โƒ— . ๐’‚ย โƒ— + ๐‘ย โƒ— . ๐‘ย โƒ— = ๐‘Žย โƒ— . ๐‘Žย โƒ— + ๐‘Žย โƒ— . ๐‘ย โƒ— + ๐’‚ย โƒ— . ๐’ƒย โƒ— + ๐‘ย โƒ— . ๐‘ย โƒ— = ๐’‚ย โƒ— . ๐’‚ย โƒ— + 2๐‘Žย โƒ— . ๐‘ย โƒ— + ๐’ƒย โƒ— . ๐’ƒย โƒ— =|๐’‚ย โƒ—|2 + 2๐‘Žย โƒ— . ๐‘ย โƒ— + |๐’ƒย โƒ—|2 Since ๐‘Žย โƒ— and ๐‘ย โƒ— are perpendicular, ๐‘Žย โƒ— . ๐‘ย โƒ— = 0 Putting ๐‘Žย โƒ— . ๐‘ย โƒ— = 0 in (1) (๐‘Žย โƒ— + ๐‘ย โƒ—) . (๐‘Žย โƒ— + ๐‘ย โƒ—) = |๐‘Žย โƒ—|2 + 2.(0) + |๐‘ย โƒ—|2 = |๐‘Žย โƒ—|2 + |๐‘ย โƒ—|2 Hence proved

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