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JEE MainMathematicsVector Algebra

Let a vector r lie in the plane of the vectors a = i + j and b = j + k . It is given that r is perpendicular to a , makes an angle of 6 with b , and has a magnitude of 6 . If c = 2 i + j - k , then the value of | r + c |^2 is equal to

Options

  1. A6
  2. B18
  3. C12
  4. D9

Correct answer

A. 6

Step-by-step solution

Since r lies in the plane of a and b , we can write r = a + b = ( i + j ) + ( j + k ) = i + ( + ) j + k . As r is perpendicular to a , we have r a = 0 . (1) + ( + )(1) + (0) = 0 2 + = 0 = -2 . Substituting , we get r = i - j - 2 k . The magnitude is given as | r | = 6 . ^2 + (- )^2 + (-2 )^2 = 6 ^2 = 6 = 1 . The angle between r and b is 6 . r b = | r | | b | ( 6 ) ( i - j - 2 k ) ( j + k ) = 6 2 3 2 - - 2 = 36 2 = 3 -3 = 3 = -1 . Thus, r = - i + j + 2 k . We are required to find | r + c |^2 where c = 2 i + j - k .

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