IIT-JEE 2009 · Paper 1, Mathematics question 26

Equality case. $|\vec a\times\vec b|\le1$ and $|\vec c\times\vec d|\le1$, so a dot product of 1 forces $\vec a\times\vec b=\vec c\times\vec d=\hat n$, a unit vector. Then $\vec a\perp\vec b$, $\vec c\perp\vec d$, and all four vectors lie in the plane perpendicular to $\hat n$, with $\vec b=\hat n\times\vec a$ and $\vec d=\hat n\times\vec c$.

Angle between $\vec b$ and $\vec d$. $\vec b\cdot\vec d=(\hat n\times\vec a)\cdot(\hat n\times\vec c)=\vec a\cdot\vec c=\frac12$, so $\vec b$ and $\vec d$ are at 60°: not parallel.

(A) and (B) fail because all four vectors are coplanar. (D) fails: $\vec d$ is $\vec c$ turned through 90° in the plane, and $\vec c$ is at 60° to $\vec a$, so $\vec a$ and $\vec d$ are at 30° or 150°, not parallel.

(C)