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GATE 2024 BT – Question 32

Engineering Mathematics · Linear Algebra: Matrices and determinants, systems of linear equations, eigenvalues and eigenvectors · 1 mark · Multiple choice

Let $\vec{OR}$ be the vector that is perpendicular to the vectors $\vec{OP} = 2\hat i - 3\hat j + \hat k$ and $\vec{OQ} = -2\hat i + \hat j + \hat k$. If the length of the vector $\vec{OR}$ is $\alpha\sqrt3$, then $\alpha$ is ________.

  1. 3
  2. 4
  3. 5
  4. 6

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Correct answer: (B) 4

Explanation

Here $\vec{OR}$ is proportional to $\vec{OP}\times\vec{OQ} = (-3-1)\hat i - (2+2)\hat j + (2-6)\hat k = -4(\hat i + \hat j + \hat k)$, which has length $4\sqrt3$, so $\alpha = 4$ (taking the cross product itself, since no other length is fixed).