JEE Advanced · 2007 · 08 Apr 2007 · Paper 2 · Vector Algebra

Since $\vec{a},\vec{b},\vec{c}$ are unit vectors and $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$, these vectors form an equilateral triangle when placed head-to-tail.

From $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$, we get $\vec{c}=-(\vec{a}+\vec{b})$.

Taking cross product with $\vec{a}$:


Similarly, $\vec{b}\times \vec{c}=\vec{a}\times \vec{b}$ and $\vec{c}\times \vec{a}=\vec{a}\times \vec{b}$.

Since the vectors are non-collinear unit vectors, $\vec{a}\times \vec{b}\ne \overrightarrow{0}$.

Therefore, $\vec{a}\times \vec{b}=\vec{b}\times \vec{c}=\vec{c}\times \vec{a}\ne \overrightarrow{0}$.