Properties of scalar product of two vectors are:
(i) The product quantity `vecA.vecB` is always a scalar. It is positive if the angle between the vectors is acute (i.e., `theta=90^(@)`) and negative if the angle between them is obtuse (i.e.,`90^(@)ltthetalt180^(@)`).
(ii) The scalar product is commutative, i.e., `vecA.vecB=vecB.vecA`
(iii) tors obey distributive law i.e., `vecA.(vecB+vecC)=vecA.vecB+vecA.vecC`
(iv) The angle between the vectors `theta=cos^(-1)[(vecA.vecB)/(AB)]`
(v) The scalar product of two vectors will be maximum when `costheta=-1,i.e.,theta=0^(@)`. when the vectors are parallel,
`(vecA.vecB)_(max)=AB`
(vi) The scalar product of two vectors will be minimum, when `costheta=-1,i.e.,theta=180^(@)(vecA.vecB)_(min)=-AB` when the vectors are anti-parallel.
(vii) If two vectors `vecAandvecB` are perpendicular to each other then their scalar product `vecA.vecB=0` because `cos90^(@)=0`. Then the vectors `vecanvecB` 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=A Acostheta=A^(2)` Here angle `theta=0^(@)`.
The magnitude or norm of the vector `vecA` is `|vecA|=Asqrt(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.hatjandhatk`,
`hati.hatj=hatj.hatk.hati=1.1cos90^(@)=0`
(xi) In terms of components the scalar product of A and B can be written as
`vecA.vecB=(A_(x)hati+A_(y)hatj+A_(z)hatk).(B_(x)hati+B_(y)hati+B_(y)hatj+B_(z)hatk)`
`=A_(x)B_(x)+A_(y)B_(y)+A_(z)B_(z),` 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 product of two vectors are: (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 `vecAandvecB`. even though the vectors `vecAandvecB` may or may not be mutually orthogonal
(ii) The vector product of two vectors is not commutative, i.e.,
`vecAxxvecBuarrvecBxxvecA`. But . `vecAxxvecB=-[vecBxxvecA]`.
Here it is worthwhile to note that `|vecAxxvecB|=|vecBxxvecA|=ABsintheta` i.e., in in the case of the product vectors `vecAxxvecBandvecBxxvecA`. the magnitudes are equal but directions are opposite to each other.
(iii) The vector product of two vectors will have maximum magnitude when `sintheta=1`, i.e., `theta=90^(@)` i.e., when the vectors `vecAandvecB` are orthogonal to each other.
`(vecAxxvecB)_(max)=ABhatn`
(iv) The vector product of two non-zero vectors will be minimum when `|sintheta|=0`, i.e., `theta=0^(@)` 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=A Asin0^(@)hatn=vec0`
In physics the null vector `vec0` is simply denoted as zero
(vi) The self-vector products of unit vectors are thus zero.
`hatixxhati=hatjxxhatj=hatkxxhatk=0`
(vii) In the case of orthogonal unit vectors, `hati,hatj,hatk`, in accordance with the night hand screw rule
`hatixxhatj=hatk,hatjxxhatk=hatiandhatkxxhati=hatj`
Also, since the cross product is not commutative,
`hatjxxhati=-hatk,hatkxxhatj=-hatiandhatixxhatk=-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 `j^(th)` component the order of multiplication is different than `hati^(th)` and `hatk^(th)` components.
(ix) If two vectors `vecA` and `vecB` form adjacent sides in a parallelogram, then the magnitude of `vecAxxvecB` will give the area of the parallelogram as represented graphically in figure.
(x) Since we can divide a parallelogram into two equal triangles as shown in the figure, the area of a triangle with `vecAandvecB` as sides is `(1)/(2)|vecAxxvecB|`. This is shown in the Figure.
A number of quantities used in Physics are defined through vector products. Particularly physical quantities representing rotational effects like torque, angular momentum, are defined through vector products.
