GATE CS 1995 Q2.13 — Linear Algebra: Matrices, Determinants, System of Linear Equations

A unit vector perpendicular to both the vectors $a=2i-3j+k$ and $b=i+j-2k$ is:

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Answer: A

A vector perpendicular to both $\vec{a}$ and $\vec{b}$ is given by their cross product $\vec{a} \times \vec{b}$ :

\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -3 & 1 \\ 1 & 1 & -2 \end{vmatrix}

Expanding the determinant:

\vec{a} \times \vec{b} = \hat{i}(6 - 1) - \hat{j}(-4 - 1) + \hat{k}(2 - (-3)) = 5\hat{i} + 5\hat{j} + 5\hat{k} = 5(\hat{i} + \hat{j} + \hat{k})

To find the unit vector, we divide this vector by its magnitude:

\text{Magnitude} = \sqrt{5^2 + 5^2 + 5^2} = 5\sqrt{3}

\text{Unit Vector} = \frac{5(\hat{i} + \hat{j} + \hat{k})}{5\sqrt{3}} = \frac{1}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k})

Thus, the correct option is A.

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Published 10 Oct 2026

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