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Ex 10.3, 11 - Show |a|b + |b|a is perpendicular to |a|b - |b|a - Scalar product - Solving

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  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise
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Ex 10.3, 11 Show that 𝑎﷯﷯ 𝑏﷯+ 𝑏﷯﷯ 𝑎﷯ is perpendicular to 𝑎﷯﷯ 𝑏﷯− 𝑏﷯﷯ 𝑎﷯, for any two nonzero vectors 𝑎﷯ and 𝑏﷯ If two vectors 𝑝﷯ and 𝑞﷯ are perpendicular to each other , then their scalar (dot) product is zero, i.e. 𝒑﷯ . 𝒒﷯ = 0 Hence, to show ( 𝑎﷯﷯ 𝑏﷯ + 𝑏﷯﷯ 𝑎﷯) is perpendicular to ( 𝑎﷯﷯ 𝑏﷯ − 𝑏﷯﷯ 𝑎﷯), We need to prove ( 𝑎﷯﷯ 𝑏﷯ + 𝑏﷯﷯ 𝑎﷯) . ( 𝑎﷯﷯ 𝑏﷯ − 𝑏﷯﷯ 𝑎﷯) = 0 Solving L.HS. ( 𝑎﷯﷯ 𝑏﷯ + 𝑏﷯﷯ 𝑎﷯) . ( 𝑎﷯﷯ 𝑏﷯ − 𝑏﷯﷯ 𝑎﷯) = 𝑎﷯﷯ 𝑏﷯﷯ . 𝑎﷯﷯ 𝑏﷯﷯ − 𝑎﷯﷯ 𝑏﷯﷯ . 𝑏﷯﷯ 𝑎﷯﷯ + 𝑏﷯﷯ 𝑎﷯﷯. 𝑎﷯﷯ 𝑏﷯﷯ – 𝑏﷯﷯ 𝑎﷯﷯ . 𝑏﷯﷯ 𝑎﷯﷯ = 𝑎﷯﷯2 𝑏﷯ . 𝑏﷯ − 𝑎﷯﷯ 𝑏﷯﷯ 𝒃﷯ . 𝒂﷯ + 𝑏﷯﷯ 𝑎﷯﷯ 𝑎﷯. 𝑏﷯ − 𝑏﷯﷯2 𝑎﷯ . 𝑎﷯ = 𝑎﷯﷯2 𝑏﷯ . 𝑏﷯ − 𝑎﷯﷯ 𝑏﷯﷯ 𝒂﷯ . 𝒃﷯ + 𝑎﷯﷯ 𝑏﷯﷯ 𝑎﷯. 𝑏﷯ − 𝑏﷯﷯2 𝑎﷯ . 𝑎﷯ = 𝑎﷯﷯2 𝒃﷯ . 𝒃﷯ − 𝑏﷯﷯2 𝒂﷯ . 𝒂﷯ = 𝑎﷯﷯2 𝒃﷯﷯𝟐 − 𝑏﷯﷯2 𝒂﷯﷯𝟐 = 0 = RHS Hence proved.

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