JEE Advanced · 2009 · 12 Apr 2009 · Paper 1 · Vector Algebra

Since $(\vec{a}\times \vec{b})\cdot (\vec{c}\times \vec{d})=1$ and all vectors are unit vectors, we need $|\vec{a}\times \vec{b}|=|\vec{c}\times \vec{d}|=1$ and $(\vec{a}\times \vec{b})\parallel (\vec{c}\times \vec{d})$.

This means $\vec{a}\perp \vec{b}$ and $\vec{c}\perp \vec{d}$, and the two cross products are parallel.

Given $\vec{a}\cdot \vec{c}=\frac{1}{2}$, we have $\cos \theta =\frac{1}{2}$, so $\theta =60°$.

If $\vec{b}\parallel \vec{d}$, then $(\vec{a}\times \vec{b})\parallel (\vec{c}\times \vec{d})$ would require specific angle relationships that contradict $\vec{a}\cdot \vec{c}=\frac{1}{2}$. Therefore, $\vec{b}$ and $\vec{d}$ are non-parallel.