Why only two kinds of vector products defined?

[Sunny Singh]

Why only the dot and cross product for vector multiplication is defined and not another kind of vector product like one in which the magnitude is that of the dot product and direction is of the cross product. I mean if A and B be two vectors then let there be a vector product such that A#B=ABcos@i where @ is the angle between A and B and i is the direction perpendicular to both of them described by right hand screw rule. Why don't we encounter any situation in Physics where such a vector product is required?? Why is the dot and cross product sufficient?

 

[A.T.]

Why only the dot and cross product for vector multiplication is defined and not another kind of vector product...

You have just defined another kind of vector product. So your question is now obsolete.

 

[Sunny Singh]

You have just defined another kind of vector product. So your question is now obsolete.

sorry my question was incomplete at that time. i want to ask why only dot and cross product is required in physics like when calculating work done and Torque? why any other type of vector product is not required for any physical situation and hence not defined?

 

[A.T.]

why any other type of vector product is not required for any physical situation and hence not defined?

There are other products defined:
http://mathworld.wolfram.com/VectorMultiplication.html

 

[dextercioby]

There are other products defined:
http://mathworld.wolfram.com/VectorMultiplication.html

Maybe they should have put the Lie product as well.

 

[Nugatory]

sorry my question was incomplete at that time. i want to ask why only dot and cross product is required in physics like when calculating work done and Torque? why any other type of vector product is not required for any physical situation and hence not defined?

Those two are the ones that have been found to be most useful (in the sense of being good tools for solving problems) in all the physics that you'll be studying before you to get to quantum mechanics. They're also needed for much of engineering and mechanical design. Thus, they're the ones that get taught first and most widely.

As other posters in this thread have pointed out, there are other possible vector products as well - most people just never never encounter them because they never study the sorts of problems in which they would be useful.