Write the important properties of vector (cross) product.

Explanation

Properties of Vector Products :
(i) The vector product of any two vectors is always another vector whose direction is perpendicular to the plane containing these two vectors, i.e., orthogonal to both the vectors, `vecA and vecB` may or may not be mutually orthogonal.
(ii) The vector product of two vectors is not commutative, `vecA xx vecB ne vecB xx vecA` But, `vecA xx vecB=-[vecBxxvecA]`. Here it is worthwhile to note that `|vecAxxvecB|=|vecBxxvecA|=AB sin theta`. i.e. in case of the product vectors `vecAxxvecB` and `vecB xx vecA` the magnitudes are equal but directions are opposite to each other.
(iii) The vector product of two vectors will have maximum magnitude when `sin theta=1`, i.e., `theta=90^(@)` or `180^(@)`
`[vecAxxvecB]_("min")=0`
i.e., the vector product of two non - zero vectors vanishes, if the vectors are either parallel or antiparallel.
(v) The self - cross product, i.e., product of a vector with itself is the null vector
`vecAxxvecA="AA "sin 0^(@)=hatn=vec0`
In physics the null vector is `vec0` simply denoted as zero.
(vi) The self - vector products of unit vectors are thus zero.
`hatixxhati=hatjxxhatj=hatkxxhatk=vec0`
(vii) In the case of orthogonal unit vectors, `hati, hatj, hatk` in accordance with the right hand screw rule :
`hatixxhatj=hatk, hatjxxhatk=hati and hatk xx hati=hatj`

Also, since the cross product is not commutative.
`hatjxxhati=-hatkxxhatj=-hati and hati xx hatk =-hatj`
(viii) In terms of components, the vector product of two vectors `vecA and vecB` is
`vecAxxvecB=|(hati, hatj, hatk),(A_(x),A_(y),A_(z)),(B_(x),B_(y),B_(z))|`
`=hati(A_(y)B_(z)-A_(z)B_(y))+hatj(A_(z)B_(x)-A_(x)B_(z))+hatk(A_(x)B_(y)-A_(y)B_(x))`
Note that in the `hatj`th component the order of multiplication is different than `hati`th and `hatk`th components.
(ix) If two vectors `vecA and vecB` from adjacent sides in a parallelogram, then the magnitude of `vecAxxvecB` will give the area of the parallelogram as represented graphically.

(x) Divide a parallelogram into two equal triangles the area of a triangle with `vecA` and `vecB` as sides is `(1)/(2)|vecAxxvecB|`.

A number of quantities used in physics are defined through vectors products, Particularly physical quantities representing rotational effect likes torque, angular momentum, are defined through vector products.