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Let |a| = 3, |b| = 4, |c| = 5 and each one being perpendicular to sum

Example 28 - Chapter 10 Class 12 Vector Algebra - Part 2
Example 28 - Chapter 10 Class 12 Vector Algebra - Part 3

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Example 28 Let π‘Ž βƒ—, 𝑏 βƒ— and 𝑐 βƒ— be three vectors such that |π‘Ž βƒ—|= 3, |𝑏 βƒ—|= 4, |𝑐 βƒ—|= 5 and each one of them being perpendicular to the sum of the other two, find |π‘Ž βƒ—" + " 𝑏 βƒ—" + " 𝑐 βƒ— |.Given, |π‘Ž βƒ—| = 3 , |𝑏 βƒ—| = 4 , |𝑐 βƒ—|= 5 Also, Each one of them being perpendicular to the sum of the other two 𝒂 βƒ— is perpendicular to 𝒃 βƒ— + 𝒄 βƒ— π‘Ž βƒ—. (𝑏 βƒ— + 𝑐 βƒ—) = 0 𝒂 βƒ—. 𝒃 βƒ— + 𝒂 βƒ—. 𝒄 βƒ— = 0 𝒃 βƒ— is perpendicular to 𝒂 βƒ— + 𝒄 βƒ— 𝑏 βƒ—. (π‘Ž βƒ— + 𝑐 βƒ—) = 0 𝒃 βƒ—. 𝒂 βƒ— + 𝒃 βƒ—. 𝒄 βƒ— = 0 𝒄 βƒ— is perpendicular to 𝒂 βƒ— + 𝒃 βƒ— 𝑐 βƒ—. (π‘Ž βƒ— + 𝑏 βƒ—) = 0 𝒄 βƒ—. 𝒂 βƒ— + 𝒄 βƒ—. 𝒃 βƒ— = 0 Now, |𝒂 βƒ—+𝒃 βƒ—+𝒄 βƒ— |2 = (𝒂 βƒ— + 𝒃 βƒ— + 𝒄 βƒ—) . (𝒂 βƒ— + 𝒃 βƒ— + 𝒄 βƒ—) = π‘Ž βƒ—. π‘Ž βƒ— + π‘Ž βƒ— . 𝑏 βƒ— + π‘Ž βƒ— . 𝑐 βƒ— + 𝑏 βƒ— . π‘Ž βƒ— + 𝑏 βƒ— . 𝑏 βƒ— + 𝑏 βƒ— . 𝑐 βƒ— + 𝑐 βƒ— . π‘Ž βƒ— + 𝑐 βƒ— . 𝑏 βƒ— + 𝑐 βƒ— . 𝑐 βƒ— = π‘Ž βƒ—. π‘Ž βƒ— + (𝒂 βƒ— . 𝒃 βƒ— + 𝒂 βƒ— . 𝒄 βƒ—) + 𝑏 βƒ— . 𝑏 βƒ— + (𝒃 βƒ— . 𝒂 βƒ— + 𝒃 βƒ— . 𝒄 βƒ—) + 𝑐 βƒ— . 𝑐 βƒ— + (𝒄 βƒ— . 𝒂 βƒ— + 𝒄 βƒ— . 𝒃 βƒ— = π‘Ž βƒ—. π‘Ž βƒ— + (0) + 𝑏 βƒ— . 𝑏 βƒ— + (0) + 𝑐 βƒ— . 𝑐 βƒ— + (0) = 𝒂 βƒ—. 𝒂 βƒ— + 𝑏 βƒ—. 𝑏 βƒ— + 𝑐 βƒ—. 𝑐 βƒ— = "|" 𝒂 βƒ—"|"2 + "|" 𝑏 βƒ—"|"2 + "|" 𝑐 βƒ—"|"2 = 32 + 42 + 52 = 9 + 16 + 25 = 50 So, |𝒂 βƒ—+𝒃 βƒ—+𝒄 βƒ— |2 = 50 Taking square root both sides, "|" π‘Ž βƒ— "+ " 𝑏 βƒ—" + " 𝑐 βƒ—"|" = √50 "|" π‘Ž βƒ— "+ " 𝑏 βƒ— "+ " 𝑐 βƒ—"|" = √25 Γ— √2 "|" 𝒂 βƒ— "+ " 𝒃 βƒ— "+ " 𝒄 βƒ—"|" = πŸ“βˆšπŸ = π‘Ž βƒ—. π‘Ž βƒ— + (0) + 𝑏 βƒ— . 𝑏 βƒ— + (0) + 𝑐 βƒ— . 𝑐 βƒ— + (0) = 𝒂 βƒ—. 𝒂 βƒ— + 𝑏 βƒ—. 𝑏 βƒ— + 𝑐 βƒ—. 𝑐 βƒ— = "|" 𝒂 βƒ—"|"2 + "|" 𝑏 βƒ—"|"2 + "|" 𝑐 βƒ—"|"2 = 32 + 42 + 52 = 9 + 16 + 25 = 50 So, |𝒂 βƒ—+𝒃 βƒ—+𝒄 βƒ— |2 = 50 Taking square root both sides, "|" π‘Ž βƒ— "+ " 𝑏 βƒ—" + " 𝑐 βƒ—"|" = √50 "|" π‘Ž βƒ— "+ " 𝑏 βƒ— "+ " 𝑐 βƒ—"|" = √25 Γ— √2 "|" 𝒂 βƒ— "+ " 𝒃 βƒ— "+ " 𝒄 βƒ—"|" = πŸ“βˆšπŸ

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