Misc 14 - If a,b,c are mutually perpendicular vectors of equal - Miscellaneous

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  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise
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Misc 14 If 𝑎﷯, 𝑏﷯, 𝑐﷯ are mutually perpendicular vectors of equal magnitudes, show that the vector 𝑎﷯ + 𝑏﷯ + 𝑐﷯ is equally inclined to 𝑎﷯, 𝑏﷯ and 𝑐﷯ . 𝑎﷯, 𝑏﷯, 𝑐﷯ are of equal magnitudes, So, 𝑎﷯﷯ = 𝑏﷯﷯ = 𝑐﷯﷯ Also, 𝑎﷯ , 𝑏﷯ , 𝑐﷯ are mutually perpendicular to each other So, 𝒂﷯ . 𝒃﷯ = 𝒃﷯ . 𝒄﷯ = 𝒄﷯ . 𝒂﷯ = 0 We need to show ( 𝑎﷯ + 𝑏﷯ + 𝑐﷯) is equally inclined to 𝑎﷯, 𝑏﷯, 𝑐﷯ ; So, cos 𝛼 = 𝑎﷯﷯﷮ 𝑎﷯ + 𝑏﷯ + 𝑐﷯﷯﷯ , cos 𝛽 = 𝑏﷯﷯﷮ 𝑎 ﷯ + 𝑏﷯ + 𝑐﷯﷯﷯ , cos 𝛾 = 𝑐﷯﷯﷮ 𝑎﷯ + 𝑏﷯ + 𝑐﷯﷯﷯ But 𝒂﷯﷯ = 𝒃﷯﷯ = 𝒄﷯﷯ Thus, cos 𝜶 = cos 𝜷= cos 𝜸 ∴ 𝛼 = 𝛽 = 𝛾 Therefore, ( 𝑎﷯ + 𝑏﷯ + 𝑐﷯) is equally inclined to 𝑎﷯, 𝑏﷯, 𝑐﷯.

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