Define vector product. Explain the properties of a vector product with two examples.

Explanation

Vector product (or) cross product: If the product of two vectors say `barA " and " bar B` gives a vector then that multiplication of vectors is called cross product or vector product of vectors.
Mathematically `barA xx barB = | barA| |barB| sin theta . hatn`
where `hatn = a` unit vector perpendicular to both `barA " and " barB` (or) perpendicular to the plane of `barA " and " bar B`.
Properties of cross product:
1. Cross product is not commutative i.e.
`barA xx barB ne bar B xx barA " But " barA xx barB = - barB xx barA`
2. Cross product obeys distributive law
`barA xx (barB+ barC) = (barA xx barB) +(barAxx barC)`
3. If any vector is represented by the combination of `bari, barj " and " bar k` then cross product will obey right hand screw rule.
4. The product of two coplanar perpendicular unit vectors will generate a unit vector perpendicular to that plane i.e.
`bari xx barj = bark , barj xx bark = bari " and " bark xx bari = barj`
5. Cross product of parallel vectors is zero i.e.
`bari xx bari = barj xx barj = bark xx bark = 0`
Examples of cross product : 1) Torque (or) moment of force (`tau` ): It is defined as the product of force and perpendicular distance from the point of application
`therefore " Torque " tau= barr xx barF`
2) Angular momentum and angular velocity: For a rigid body in motion, Angular momentum (`barL`) = radius (`barr`) x momentum (barP`)
`therefore` Angular momentum (`barL`)
` =barr xx (mbarv) = m(barrxx barv)`