Ex 10.3, 6 - Find |a|, |b|, if (a + b).(a - b) = 8, |a| = 8|b| - Scalar product - Solving

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  1. Chapter 10 Class 12 Vector Algebra
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Ex 10.3, 6 Find 𝑎īˇ¯īˇ¯ and 𝑏īˇ¯īˇ¯, if ( 𝑎īˇ¯ + bīˇ¯) .( 𝑎īˇ¯ – 𝑏īˇ¯) = 8 and 𝑎īˇ¯īˇ¯ = 8 𝑏īˇ¯īˇ¯ ( 𝑎īˇ¯ + bīˇ¯) . ( 𝑎īˇ¯ – 𝑏īˇ¯) = 8 𝑎īˇ¯.( 𝑎īˇ¯ – bīˇ¯) + 𝑏īˇ¯.( 𝑎īˇ¯ − 𝑏īˇ¯) = 8 𝑎īˇ¯. 𝑎īˇ¯ − 𝑎īˇ¯. 𝑏īˇ¯ + 𝒃īˇ¯. 𝒂īˇ¯ − 𝑏īˇ¯. 𝑏īˇ¯ = 8 𝑎īˇ¯. 𝑎īˇ¯ − 𝑎īˇ¯. 𝑏īˇ¯ + 𝒂īˇ¯. 𝒃īˇ¯ − 𝑏īˇ¯. 𝑏īˇ¯ = 8 𝒂īˇ¯. 𝒂īˇ¯ − 𝒃īˇ¯. 𝒃īˇ¯ = 8 𝒂īˇ¯īˇ¯2 – 𝒃īˇ¯īˇ¯2 = 8 Also given that , 𝑎īˇ¯īˇ¯ = 8 𝑏īˇ¯īˇ¯ Putting this in (1), 𝒂īˇ¯īˇ¯2 – 𝑏īˇ¯īˇ¯2 = 8 (8 𝒃īˇ¯īˇ¯)2 − 𝑏īˇ¯īˇ¯2 = 8 64 𝑏īˇ¯īˇ¯2 − 𝑏īˇ¯īˇ¯2 = 8 63 𝑏īˇ¯īˇ¯2 = 8 𝑏īˇ¯īˇ¯2 = 8īˇŽ63īˇ¯ 𝑏īˇ¯īˇ¯ = īˇŽ8īˇ¯īˇŽ īˇŽ63īˇ¯īˇ¯ ∴ 𝒃īˇ¯īˇ¯ = 𝟐 īˇŽđŸīˇ¯īˇŽđŸ‘ īˇŽđŸ•īˇ¯īˇ¯ Also, 𝑎īˇ¯īˇ¯ = 8 × 𝑏īˇ¯īˇ¯ 𝑎īˇ¯īˇ¯ = 8 × 2 īˇŽ2īˇ¯īˇŽ3 īˇŽ7īˇ¯īˇ¯ ∴ 𝒂īˇ¯īˇ¯ = 𝟏𝟔 īˇŽđŸīˇ¯īˇŽđŸ‘ īˇŽđŸ•īˇ¯īˇ¯

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