Example 28 - Chapter 10 Class 12 Vector Algebra
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
Example 28 Let 𝑎 ⃗, 𝑏 ⃗ and 𝑐 ⃗ be three vectors such that |𝑎 ⃗|= 3, |𝑏 ⃗|= 4, |𝑐 ⃗|= 5 and each one of them being perpendicular to the sum of the other two, find |𝑎 ⃗" + " 𝑏 ⃗" + " 𝑐 ⃗ |.Given, |𝑎 ⃗| = 3 , |𝑏 ⃗| = 4 , |𝑐 ⃗|= 5 Also, Each one of them being perpendicular to the sum of the other two 𝒂 ⃗ is perpendicular to 𝒃 ⃗ + 𝒄 ⃗ 𝑎 ⃗. (𝑏 ⃗ + 𝑐 ⃗) = 0 𝒂 ⃗. 𝒃 ⃗ + 𝒂 ⃗. 𝒄 ⃗ = 0 𝒃 ⃗ is perpendicular to 𝒂 ⃗ + 𝒄 ⃗ 𝑏 ⃗. (𝑎 ⃗ + 𝑐 ⃗) = 0 𝒃 ⃗. 𝒂 ⃗ + 𝒃 ⃗. 𝒄 ⃗ = 0 𝒄 ⃗ is perpendicular to 𝒂 ⃗ + 𝒃 ⃗ 𝑐 ⃗. (𝑎 ⃗ + 𝑏 ⃗) = 0 𝒄 ⃗. 𝒂 ⃗ + 𝒄 ⃗. 𝒃 ⃗ = 0 Now, |𝒂 ⃗+𝒃 ⃗+𝒄 ⃗ |2 = (𝒂 ⃗ + 𝒃 ⃗ + 𝒄 ⃗) . (𝒂 ⃗ + 𝒃 ⃗ + 𝒄 ⃗) = 𝑎 ⃗. 𝑎 ⃗ + 𝑎 ⃗ . 𝑏 ⃗ + 𝑎 ⃗ . 𝑐 ⃗ + 𝑏 ⃗ . 𝑎 ⃗ + 𝑏 ⃗ . 𝑏 ⃗ + 𝑏 ⃗ . 𝑐 ⃗ + 𝑐 ⃗ . 𝑎 ⃗ + 𝑐 ⃗ . 𝑏 ⃗ + 𝑐 ⃗ . 𝑐 ⃗ = 𝑎 ⃗. 𝑎 ⃗ + (𝒂 ⃗ . 𝒃 ⃗ + 𝒂 ⃗ . 𝒄 ⃗) + 𝑏 ⃗ . 𝑏 ⃗ + (𝒃 ⃗ . 𝒂 ⃗ + 𝒃 ⃗ . 𝒄 ⃗) + 𝑐 ⃗ . 𝑐 ⃗ + (𝒄 ⃗ . 𝒂 ⃗ + 𝒄 ⃗ . 𝒃 ⃗ = 𝑎 ⃗. 𝑎 ⃗ + (0) + 𝑏 ⃗ . 𝑏 ⃗ + (0) + 𝑐 ⃗ . 𝑐 ⃗ + (0) = 𝒂 ⃗. 𝒂 ⃗ + 𝑏 ⃗. 𝑏 ⃗ + 𝑐 ⃗. 𝑐 ⃗ = "|" 𝒂 ⃗"|"2 + "|" 𝑏 ⃗"|"2 + "|" 𝑐 ⃗"|"2 = 32 + 42 + 52 = 9 + 16 + 25 = 50 So, |𝒂 ⃗+𝒃 ⃗+𝒄 ⃗ |2 = 50 Taking square root both sides, "|" 𝑎 ⃗ "+ " 𝑏 ⃗" + " 𝑐 ⃗"|" = √50 "|" 𝑎 ⃗ "+ " 𝑏 ⃗ "+ " 𝑐 ⃗"|" = √25 × √2 "|" 𝒂 ⃗ "+ " 𝒃 ⃗ "+ " 𝒄 ⃗"|" = 𝟓√𝟐 = 𝑎 ⃗. 𝑎 ⃗ + (0) + 𝑏 ⃗ . 𝑏 ⃗ + (0) + 𝑐 ⃗ . 𝑐 ⃗ + (0) = 𝒂 ⃗. 𝒂 ⃗ + 𝑏 ⃗. 𝑏 ⃗ + 𝑐 ⃗. 𝑐 ⃗ = "|" 𝒂 ⃗"|"2 + "|" 𝑏 ⃗"|"2 + "|" 𝑐 ⃗"|"2 = 32 + 42 + 52 = 9 + 16 + 25 = 50 So, |𝒂 ⃗+𝒃 ⃗+𝒄 ⃗ |2 = 50 Taking square root both sides, "|" 𝑎 ⃗ "+ " 𝑏 ⃗" + " 𝑐 ⃗"|" = √50 "|" 𝑎 ⃗ "+ " 𝑏 ⃗ "+ " 𝑐 ⃗"|" = √25 × √2 "|" 𝒂 ⃗ "+ " 𝒃 ⃗ "+ " 𝒄 ⃗"|" = 𝟓√𝟐
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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo