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Vector Product of Two Vectors

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  • Definition of Vector Product

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Vector Product or Cross Product of two vectors:

The vector product or cros product of two vectors `vec"A"` and `vec"B"` is another vector `vec"C"`, whose magnitude is equal to the product of the magnitudes of the two vectors and sine of the smaller angle between them.

If Θ is the smaller angle between `vec"A"` and `vec"B"`, then

`vec"A"xxvec"B" = vec"C" = "AB" sin theta hat"C"`

where `hatC` is a unit vector in the direction of `vecC`. The direction of `vecC` or `hatC`(i.e. vector product of two vectors) is perpendicular to the plane containing `vec"A"` and `vec"B"` and pointing in the direction of advance of a right handed screw when rotated from `vec"A"` to `vec"B"`.

Some important properties of cross products are as follows:

(a) For parallel as well as anti parallel vectors(i.e. when `theta` = 0° or 180°), the cross product is zero.

(b) the magnitude of cross product of two perpendicular vectors is equal to the product of the magnitudes of the given vectors.

(c) Vector product is anti-commutative i.e. `vec"A" xx vec"E" = -vec"B" xx vec"A"` 

(d) Vector product is distributive i.e. `vec"A"xx(vec"B" +vec"C")=vec"A" xx vec"B" + vecA xx vec"C"`

(e)`vecA xx vecB` does not change sign under reflection i.e. `(-vecA)xx(-vecB)=vecA xx vecB`

(f) For unit orthogonal vectors, we have

`hatixxhati=hatjxxhatj=hatkxxhatk=0,hatixxhatj=hatk,hatjxxhatk=hati and hatkxxhati=hatj`

moreover `hatjxxhati=-hatk, hatkxxhatj=-hati and hatixxhatk=-hatj`

(g) In terms of components `vec"A"xxvec"B"= |(hati,hatj,hatk),(A_x,A_y,A_z),(B_x,B_y,B_z)|`

The angular velocity of a body or a particle is defined as the ratio of the angular dispacement of the body or the particle to the time interval during which this displacement occurs.

ω = `("d" theta)/"dt"`

The direction of angular velocity is along the axis of rotation. it is measured in radian/sec and its dimensional formula is [M°L°T-1].

The relation betwennt angular velocity and linear velocity is given by

`vecv = vecomegaxxvecr`

The angular acceleration of a body is defined as the ratio of the change in the angular velocity to the time interval.

`"Angular acceleration" = "Change in angular velocity"/("Time taken")`

`vecalpha = ("d"vecomega)/"dt"`

The unit of angular acceleration is rad s-2 and dimensional formula is [M°L°L-2]

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