# 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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