UPSC NDA I 2026 Mathematics Paper

Concept:A cross product vector is perpendicular to each of the two vectors being multiplied. This gives orthogonality, but unit length is not necessarily forced.Explanation:Given $\vec c = \vec a \times \vec b$, we get $\vec c \perp \vec a$ and $\vec c \perp \vec b$.Given $\vec a = \vec b \times \vec c$, we get $\vec a \perp \vec b$ and $\vec a \perp \vec c$.Hence $\vec a, \vec b, \vec c$ are orthogonal in pairs, so Statement I is correct.But their magnitudes need not all be 1. For example, let $\vec a = 2\mathbf i$, $\vec b = \mathbf j$, $\vec c = 2\mathbf k$. Then:$$\vec a \times \vec b = 2\mathbf k = \vec c, \quad \vec b \times \vec c = 2\mathbf i = \vec a$$Here $\vec a$ and $\vec c$ are not unit vectors. So Statement II is incorrect.Answer:I only, i.e. Option A.