Q.Find the area of the parallelogram whose adjacent sides are $\vec a = 3\hat i + \hat j + 4\hat k$ and $\vec b = \hat i - \hat j + \hat k$.

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. …