Q.Show that $\vec a\times(\vec b+\vec c)+\vec b\times(\vec c+\vec a)+\vec c\times(\vec a+\vec b)=\vec 0$.

Vector (Cross) Product

The vector product of two vectors $\vec A$ and $\vec B$ is a vector

whose magnitude is $AB\sin\theta$ ($\theta$ the angle between them) and whose direction $\hat n$ is perpendicular to the plane of $\vec A$ and $\vec B$, given by the right‑hand rule.

Key properties: it is anti‑commutative, $\vec A\times\vec B = -\,\vec B\times\vec A$, so the two products point in opposite directions (angle $180^\circ$ between them); the cross product of parallel or anti‑parallel vectors is the null vector ($\sin 0 = 0$); and it is maximum for perpendicular vectors. In components,

Dividing $\vec A\times\vec B$ by its magnitude gives the unit vector perpendicular to both. The cross product defines torque, angular momentum, magnetic force and area vectors.

The vector (cross) product is foundational to both the Class 11-12 Mathematics vectors chapter and the Class 11 Physics treatment of torque and angular momentum, commonly searched as "vector cross product formula and examples" or "cross product important questions class 12". Its anti-commutative property and use in finding a unit vector perpendicular to two given vectors are frequently tested in board exams and JEE Main.