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JEE MainMathematicsVector Algebra

Let a = i + 2 j + 2 k and b = 2 i - 2 j + k . If c is a vector such that ( a + b ) c = 2( c a ) for some scalar , and it is given that c a = 54 and c b = 90 , then the value of is equal to _______

Correct answer

5

Step-by-step solution

Given the vector equation: ( a + b ) c = 2( c a ) Using the anti-commutative property of the cross product, 2( c a ) = -2( a c ) . Substituting this into the equation, we get: ( a + b ) c + 2( a c ) = 0 Using the distributive property: ( a + b + 2 a ) c = 0 (3 a + b ) c = 0 This implies that c is collinear with (3 a + b ) , so we can write: c = (3 a + b ) for some scalar . Now, we find the dot products and magnitudes for a and b : | a |^2 = 1^2 + 2^2 + 2^2 = 9 | b |^2 = 2^2 + (-2)^2 + 1^2 = 9 a b = (1)(2) + (2)(-2)

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