Misc 4 - If a = b + c, then is it true that |a| = |b| + |c|

Misc 4 - Chapter 10 Class 12 Vector Algebra - Part 2
Misc 4 - Chapter 10 Class 12 Vector Algebra - Part 3

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Transcript

Misc 4 If 𝑎 ⃗ = 𝑏 ⃗ + 𝑐 ⃗ , then is it true that |𝑎 ⃗|=|𝑏 ⃗| + |𝑐 ⃗| Justify your answer.Given, 𝒂 ⃗ = 𝒃 ⃗ + 𝒄 ⃗ Let 𝒃 ⃗ = 1𝑖 ̂ + 2𝑗 ̂ + 3𝑘 ̂ & 𝒄 ⃗ = 2𝑖 ̂ − 1𝑗 ̂ − 2𝑘 ̂ Thus, 𝒂 ⃗ = (𝑏 ⃗ + 𝑐 ⃗) = (1 + 2) 𝑖 ̂ + (2 − 1) 𝑗 ̂ + (3 − 2) 𝑘 ̂ = 3𝒊 ̂ + 1𝒋 ̂ + 1𝒌 ̂ Given, 𝒂 ⃗ = 𝒃 ⃗ + 𝒄 ⃗ Let 𝒃 ⃗ = 1𝑖 ̂ + 2𝑗 ̂ + 3𝑘 ̂ & 𝒄 ⃗ = 2𝑖 ̂ − 1𝑗 ̂ − 2𝑘 ̂ Thus, 𝒂 ⃗ = (𝑏 ⃗ + 𝑐 ⃗) = (1 + 2) 𝑖 ̂ + (2 − 1) 𝑗 ̂ + (3 − 2) 𝑘 ̂ = 3𝒊 ̂ + 1𝒋 ̂ + 1𝒌 ̂ Given, 𝒂 ⃗ = 𝒃 ⃗ + 𝒄 ⃗ Let 𝒃 ⃗ = 1𝑖 ̂ + 2𝑗 ̂ + 3𝑘 ̂ & 𝒄 ⃗ = 2𝑖 ̂ − 1𝑗 ̂ − 2𝑘 ̂ Thus, 𝒂 ⃗ = (𝑏 ⃗ + 𝑐 ⃗) = (1 + 2) 𝑖 ̂ + (2 − 1) 𝑗 ̂ + (3 − 2) 𝑘 ̂ = 3𝒊 ̂ + 1𝒋 ̂ + 1𝒌 ̂ Finding |𝒂 ⃗ |, |𝒃 ⃗ | , |𝒄 ⃗ | Magnitude of 𝑎 ⃗ = √(32+1^2+1^2 ) |𝒂 ⃗ | = √(9+1+1) = √𝟏𝟏 Magnitude of 𝑏 ⃗ = √(12+22+32) |𝒃 ⃗ | = √(1+4+9) = √𝟏𝟒 Magnitude of 𝑐 ⃗ = √(22+(−1)2+(−2)2) |𝒄 ⃗ | = √(4+1+4) = √9 = 3 Now, |𝒃 ⃗ | + |𝒄 ⃗ | = √14 + 3 ≠ √11 ≠ |𝒂 ⃗ | So, |𝑎 ⃗ |≠ |𝑏 ⃗ | + |𝑐 ⃗ | Hence, the given statement is False.

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.