Properties of Scalar Products:
(i) The product quantity `vecA. vecB` is always a scalar. It is positive if the angle between the vectors is acute (i.e. `lt 90^(@)`) and negative if the angle between them is obtuse (i.e. `90^(@) lt theta lt 180^(@)`).
(ii) The scalar product is commutative,
i.e., `vecA.vecB=vecB.vecA`
(iii) The vectors obey distributive law i.e.
`vecA.(vecB+vecC)=vecA.vecB+vecA+vecB`
(iv) The angle between the vectors
`theta=cos^(-1)[(vecA.vecB)/(AB)]`
(v) The scalar product of two vectors will be maximum when `cos theta=1`, i.e., `theta =0^(@)`, i.e., when the vectors are parallel ,
`(vecA.vecB)_("max")=AB`
(vi) The scalar product of two vectors will be minimu, when `cos theta=-1`, i.e. `theta=180^(@)`
`(vecA.vecB)_("min")=-AB` when the vectors are anti - parallel.
(vii) If two vectors `vecA and vecB` are perpendicular to each other then their scalar product `vecA.vecB=0`, because `cos 90^(@)=0`. Then the vectors `vecA and vecB` are said to be mutually orthogonal.
(viii) The scalar product of a vector with itself is termed as self - dot product and is given by `(vecA)^(2)=vecA.vecA="AA "cos theta=A^(2)` Here angle `theta=0^(@)`
The magnitude or norm of the vector `vecA` is `|vecA|=A=sqrt(vecA.vecA)`
(ix) In case of a unit vector `hatn`
`hatn.hatn=1xx1xxcos0=1`. For example, `hati.hati=hatj.hatj=hatk.hatk=1`.
(x) In the case of orthogonal unit vectors `hati, hatj and hatk`.
`hati.hatj=hatj.hatk.hati=1.1 cos 90^(@)=0`
(xi) In terms of components the scalar product of `vecA and vecB` can be written as
`vecA.vecB=(A_(2)hati+A_(y)hatj+A_(z)hatk).(B_(x)hati+B_(y)hatj+B_(z)hatk)` with all other terms zero.
The magnitude of vector `|vecA|` is given by
`|vecA|=A=sqrt(A_(x)^(2)+A_(y)^(2)+A_(z)^(2))`
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.
