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JEE Main2012MathematicsThree Dimensional GeometryActual

A unit vector which is perpendicular to the vector 2 i - j +2 k and is coplanar with the vectors i + j - k and 2 i +2 j - k is

Options

  1. A2 j + k 5
  2. B3 i +2 j -2 k 17
  3. C3 i +2 j +2 k 17
  4. D2 i +2 j - k 3

Correct answer

D. 2 i +2 j - k 3

Step-by-step solution

Let x i +y j +z k be the required unit vector. Since a is perpendicular to (2 i - j +2 k ) . 2 x-y+2 z=0 Since vector x i +y j +z k is coplanar with the vector i + j - k and 2 i +2 j - k . aligned x i & +y j +z k & =p( i + j - k )+q(2 i +2 j - k ), aligned where p and q are some scalars. aligned & x i +y j +z k & =(p+2 q) i +(p+2 q) j -(p+q) k & x=p+2 q, y=p+2 q, z=-p-q aligned Now from equation (i), aligned & 2 p+4 q-p-2 q-2 p-2 q=0 & -p=0 p=0 & x=2 q, y=2 q, z=-q aligned Since vector x i +y j +z k is a unit vecto

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