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1987

JEE Main 1987 Mathematics question on Vector Algebra

The number of vectors of unit length perpendicular to vectors a⃗=(1,1,0)\vec{a} = (1, 1, 0) and b⃗=(0,1,1)\vec{b} = (0, 1, 1) is
  1. A.
    one
  2. B.
    two
  3. C.
    three
  4. D.
    infinite
Show answer and solution

Correct option: B

Solution

The cross product a × b gives one vector perpendicular to both: a × b = (1·1 - 0·1, 0·0 - 1·1, 1·1 - 1·0) = (1, -1, 1), with magnitude √3. Normalizing gives (1/√3)(1, -1, 1). Its negative -(1/√3)(1, -1, 1) is also a unit vector perpendicular to both. These are the only two unit vectors in that direction, since the perpendicular direction to two non-parallel vectors in 3D is unique (up to sign).

[1987]

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