(1) `vecaxxvecb=vecbxxveca`
The vector product of two vector is not commutative ut `vecaxxvecb=-vecbxxveca` is opposite to each other.
However `|vecaxxvecb|=|vecbxxveca|`
(2) Scalar product act behave like reflection (taking image in mirror) `xrarr-x,yrarr-yandzrarr-z`.
In reflection occurrence all components change sign mean positive vector becomes negative.
So, `vecaxxvecbrarr(-veca)xx(-vecb)=vecaxxvecb`
Hence, in reflection sign is not change in resultant.
(3) Vector product obeys distributive law :
`vecaxx(vecb+vecc)=vecaxxvecb+vecaxxvecc`
(4) For two non-zero vectors `vecaxxveca=vec0`
where `vec0` is vector of zero modulus
Here `vecaxxveca=(a)(a)sin0^(@)hatn`
`=vec0` (`because` Angle between `veca and veca` is `0^(@)`)
Hence, condition of parallel or anti parallel of two non-zero vectors is that its vector product should be zero.
(5) If two non-zero vector is perpendicular, then
`vecaxxvecb=ab sin90^(@)hatn`
`=abhatn`
where `hatn` is unit vector in direction of `veca xx vecb`.
(6) Vector product for unit vector of cartesian co-ordinate system.
In cartesian co-ordinate system `hati,hatj and hatk` are the unit vectors in the direction of X, Y and Z-axis respectively,
`therefore hatixxhati=|1||1|sin0^(@),hatn`
`=vec0`
`hatjxxhatj=|1||1|sin0^(@).hatn`
`=vec0`
and `hatkxxhatk=|1||1|sin0^(@).hatn`
`vec0`
and `hatixxhatj=|1||1|sin90^(@).hatn`
`=1hatk` [`because hatk` is perpendicular unit vector of the plane `hatixxhatj`]
`=1(hatk)`
`=hatk`
Similarly `hatjxxhatk=hatiandhatkxxhati=hatj`
and `hatjxxhati=-hatk`
`hatkxxhatj=-hatiandhatixxhatk=-hatj`
(7) Vector product of two vectors in cartesian co-ordinate system :
Suppose `veca=a_(x)hati+a_(y)hatj+a_(z)hatk` and
`vecb=b_(x)hati+b_(y)hatj+b_(z)hatk` then
`vecaxxvecb=(a_(x)hati+a_(y)hatj.+a_(z)hatk)xx(b_(x)hati+b_(y)hatj+b_(z)hatk)`
`=a_(x)b_(x)(hatixxhati)+a_(x)b_(y)(hatixxhatj)+a_(x)b_(z)(hatixxveck)`
`+a_(y)b_(x)(hatjxxhati)+a_(y)b_(y)(hatjxxhatj)+a_(y)b_(z)(hatjxxhatk)`
`+a_(z)b_(x)(hatkxxhati)+a_(z)b_(y)(hatkxxhatj)+a_(z)b_(z)(hatkxxhatk)`
`therefore` From property (6) result,
`vecaxxvecb=a_(x)b_(y)(hatk)+a_(x)b_(z)(-hatj)+a_(y)b_(x)(-hatk)+a_(y)b_(z)(hati)+a_(z)b_(x)(hatj)+a_(z)b_(y)(-hati)`
`=a_(y)b_(z)hati-a_(z)b_(y)hati+a_(z)b_(x)hatj-a_(x)b_(z)hatj+a_(x)b_(y)hatk-a_(y)b_(x)hatk`
`vecaxxvecb=|(hati,hatj,hatk),(a_(x),a_(y),a_z),(b_(x),b_(y),b_(z))|`
