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JEE MainMathematicsVector Algebra

Let a and b be two unit vectors such that the angle between them is 3 . If c is a non-zero vector such that a + 3 b is collinear with c and b + 4 c is collinear with a , then the value of 144| c |^2 is equal to

Options

  1. A16
  2. B10
  3. C7
  4. D13

Correct answer

D. 13

Step-by-step solution

Given that a + 3 b is collinear with c , we can write: a + 3 b = c (1) Also, b + 4 c is collinear with a , so: b + 4 c = a c = 4 a - 1 4 b (2) Substituting (2) into (1) gives: a + 3 b = ( 4 a - 1 4 b ) Rearranging the terms, we get: ( 1 - 4 ) a + ( 3 + 4 ) b = 0 Since a and b are non-collinear (the angle between them is 3 ), their coefficients must be zero: 3 + 4 = 0 = -12 1 - 4 = 0 1 - -12 4 = 0 = - 1 3 Using = -12 in equation (1) , we have: a + 3 b = -12 c c = - 1 12 ( a + 3 b ) Now, we find 144| c |^2 : 144| c |

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