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Ex 10.4, 8 - If either a = 0 or b = 0, then a x b = 0. Converse

Ex 10.4, 8 - Chapter 10 Class 12 Vector Algebra - Part 2

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Ex 10.4, 8 If either 𝑎 ⃗ = 0 ⃗ or 𝑏 ⃗ = 0 ⃗, then 𝑎 ⃗ × 𝑏 ⃗ = 0 ⃗ . Is the converse true? Justify your answer with an example. Converse : If 𝑎 ⃗ × 𝑏 ⃗ = 0 ⃗, then either 𝑎 ⃗ = 0 ⃗ or 𝑏 ⃗ = 0 ⃗ 𝑎 ⃗ × 𝑏 ⃗ = |𝑎 ⃗ ||𝑏 ⃗ | sin θ 𝑛 ̂ where, θ = angle between 𝑎 ⃗ and 𝑏 ⃗ 𝑛 ̂ = unit vector perpendicular to 𝑎 ⃗ 𝜀 𝑏 ⃗ Let 𝑎 ⃗ = 1𝑖 ̂ + 1𝑗 ̂ + 1𝑘 ̂ & 𝑏 ⃗ = 2𝑖 ̂ + 2𝑗 ̂ + 2𝑘 ̂ 𝑎 ⃗ × 𝑏 ⃗ = |■8(𝑖 ̂&𝑗 ̂&𝑘 ̂@1&1&1@2&2&2)| = 𝑖 ̂ (1 × 2 − 2 × 1) − 𝑗 ̂ (1 × 2 −2 × 1) + 𝑘 ̂ (1 × 2 − 2 × 1) + 𝑘 ̂(1 × 2 − 2 × 1) = 𝑖 ̂ (2 − 2) − 𝑗 ̂ (2 −2) + 𝑘 ̂ (2 − 2) = 0𝑖 ̂ − 0𝑗 ̂ + 0𝑘 ̂ = 0 ⃗ Here, 𝑎 ⃗ ≠ 0 ⃗ & 𝑏 ⃗≠ 0 ⃗ But 𝑎 ⃗ × 𝑏 ⃗ = 0 ⃗ Therefore, converse is not true.

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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.