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JEE MainMathematicsVector Algebra

Let b = i + 2 j - 3 k where Z . If a is a vector such that a b = 7 i - 5 j + k , a b = 10 and | a |^2 = 14 , then the value of ^2 - 10 is equal to

Correct answer

84

Step-by-step solution

Since a b is orthogonal to b , we have b ( a b ) = 0 . (7) + 2(-5) - 3( ) = 0 7 - 10 - 3 = 0 3 = 7 - 10 Using Lagrange's identity, | a b |^2 + ( a b )^2 = | a |^2 | b |^2 (7^2 + (-5)^2 + ^2) + 10^2 = 14( ^2 + 2^2 + (-3)^2) 74 + ^2 + 100 = 14( ^2 + 13) ^2 = 14 ^2 + 8 Substituting = 7 - 10 3 into the equation: ( 7 - 10 3 )^2 = 14 ^2 + 8 49 ^2 - 140 + 100 = 9(14 ^2 + 8) 49 ^2 - 140 + 100 = 126 ^2 + 72 77 ^2 + 140 - 28 = 0 11 ^2 + 20 - 4 = 0 (11 - 2)( + 2) = 0 Since Z , we get = -2 . Then, 3 = 7(-2) - 10 = -24 = -8 . T

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