UPSC NDA I 2025 Mathematics Paper

Concept:For two vectors, the magnitude of their cross product equals the product of their magnitudes times the sine of the angle between them. The dot product equals the product of magnitudes times the cosine of that angle.Explanation:We are given that $\vec{a}$, $\vec{b}$, and $\vec{a} \times \vec{b}$ are all unit vectors. So $|\vec{a}| = 1$, $|\vec{b}| = 1$, and $|\vec{a} \times \vec{b}| = 1$. The magnitude of the cross product is $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}| \sin\theta = 1 \cdot 1 \cdot \sin\theta = \sin\theta$. Since $|\vec{a} \times \vec{b}| = 1$, we get $\sin\theta = 1$, so $\theta = 90^\circ$ (vectors are perpendicular). Now compute the dot product: $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| \cos\theta = 1 \cdot 1 \cdot \cos 90^\circ = 0$.Answer:Option A: 0