What is the physical significance of cross product. Please explain it briefly.
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In practice we come across the situations in which we have to multiply one vector with the component of another vector on it and the result is a scalar quantity. For example consider the definition of work done by a force
F in displacing an object by displacement vector d .
The work done is defined as the product of magnitude of displacement and component of force in the direction of displacement. Here, work is scalar quantity.
If (theta) is the angle between F and d and work done is W then,
W=|d||F| cos(theta). This type of product is known as scalar product of two vectors and denoted by W= F.d. With this notation this product is also known as dot product of two vectors.
We note that in physics we have to deal with product of two vectors in which we have product of magnitudes of the vectors with sine of angle between them and the result is a vector quantity.
As an example, consider a particle moving on a circular path about an axis. If r is radius of the circle, w is angular velocity and v is linear speed along the circumference ,we can take these quantities as vectors.
w is vector along axis of rotation, the direction of which is determined as follows.
Hold a right hand screw parallel to the axis of rotation and imagine it to be rotated in the same way as the particle is rotating. Then the direction in which the screw advances is the direction of w.
The particle has local velocity at each point of circumference , in the direction of tangent at that point. This is a vector v.
Also, we define the position vector of the particle with center of the circle as origin. We denote this vector by r.
Now, we get a relation wXr=v . This product is called vector product of two vectors w and r or cross product from the notation.
Now, v perpendicular to the plane formed by w
and r. This is special case of the following definition of cross product.
AXB=|A||B| sin( theta) n, where n is unit vector perpendicular to the plane formed by A and B
according to right hand screw law.
Hold the right hand screw perpendicular to the plane formed by A and B and rotate it from A
towards B then direction of advancement of the screw gives direction of n.
Torque T= rXF
angular momentum l= rXp are other examples of cross product.
F in displacing an object by displacement vector d .
The work done is defined as the product of magnitude of displacement and component of force in the direction of displacement. Here, work is scalar quantity.
If (theta) is the angle between F and d and work done is W then,
W=|d||F| cos(theta). This type of product is known as scalar product of two vectors and denoted by W= F.d. With this notation this product is also known as dot product of two vectors.
We note that in physics we have to deal with product of two vectors in which we have product of magnitudes of the vectors with sine of angle between them and the result is a vector quantity.
As an example, consider a particle moving on a circular path about an axis. If r is radius of the circle, w is angular velocity and v is linear speed along the circumference ,we can take these quantities as vectors.
w is vector along axis of rotation, the direction of which is determined as follows.
Hold a right hand screw parallel to the axis of rotation and imagine it to be rotated in the same way as the particle is rotating. Then the direction in which the screw advances is the direction of w.
The particle has local velocity at each point of circumference , in the direction of tangent at that point. This is a vector v.
Also, we define the position vector of the particle with center of the circle as origin. We denote this vector by r.
Now, we get a relation wXr=v . This product is called vector product of two vectors w and r or cross product from the notation.
Now, v perpendicular to the plane formed by w
and r. This is special case of the following definition of cross product.
AXB=|A||B| sin( theta) n, where n is unit vector perpendicular to the plane formed by A and B
according to right hand screw law.
Hold the right hand screw perpendicular to the plane formed by A and B and rotate it from A
towards B then direction of advancement of the screw gives direction of n.
Torque T= rXF
angular momentum l= rXp are other examples of cross product.
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In mathematics and vector algebra, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space {\displaystyle \left(\mathbb {R} ^{3}\right)} {\displaystyle \left(\mathbb {R} ^{3}\right)} and is denoted by the symbol {\displaystyle \times } \times . Given two linearly independent vectors {\displaystyle \mathbf {a} } \mathbf {a} and {\displaystyle \mathbf {b} } \mathbf {b} , the cross product, {\displaystyle \mathbf {a} \times \mathbf {b} } {\displaystyle \mathbf {a} \times \mathbf {b} }, is a vector that is perpendicular to both {\displaystyle \mathbf {a} } \mathbf {a} and {\displaystyle \mathbf {b} } \mathbf {b} and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with dot product (projection product).
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