what is the literal meaning of vector product? how could two vectors perpendicular to each other form a new vector which is totally out of their plane how is that possible? can we prove it mathematically as well as logically?
Geometrically the cross product is the area enclosed by completing the parallelogram the two vectors form. Any two vectors lie in the same plane. To convince yourself of this note that a vector is not anchored to a specific point in space. The cross produce defines this plane.
I'm pretty sure a cross product is just something mathematicians invented. It's perpendicular because it was defined to be that way.
Mechanically, the cross product is a screw motion. When you turn a screw you are applying a force along a tangent of the head. The rigid material turns that into a rotation and thence a movement into the wood as the screw bites. Or a spiral staircase. You walk forward. The wall deflects you sideways and the stairs force you upwards. In both cases there are 3 vectors and a cross product involved.
There is no proof of this... Its defined this way... If there are two vectors a and b then their cross product is a vector with value absin* and with direction perpendicular to that plane... And for this definition we get to see it in many places like torque,magnetic field... If it were defined differently then we wouldn't be using it in these cases... Now u must be asking urself why sin* why not tan*... Its just defined this way... You can define sth with tan*... But thing is we use it make things easier for us... So we defined sth that would come to our use... Bently's example shows us how cross can be used to describe the whole event with just a simple cross product... We also defined dot product... That comes up a lot too... In that case product is scaler with value abcos*... Why not sin*... Cause it defined this way... U can define a scaler product with sin*... If that can be used to describe things your product system might get accepted too... :)
It's not just a mathematical 'thing' though. It's something that happens in nature in thousands of different ways. There's the screw and spiral and gyroscope and also the behaviour of charges in a B field, numerous places in QM... Screw dynamics behaviour is a fundamental part of the universe. I don't think mathematicians can claim credit for 'inventing' it.
Someone else is also discussing this question. https://www.physicsforums.com/showthread.php?p=4131823#post4131823
Planes are tough to work with. How do I define the orientation of a plane? Vectors are a lot easier to work with. So instead of working with a plane, I work with a vector that's perpendicular to the plane I'm interested in. Essentially, the cross product is doing two things. It's finding the area of a portion of plane that's bounded by the two vectors, finding the relative orientation of that plane, and referring to both by the vector that was created by the cross product. May not be easy to see if you're taking the cross product of two two-dimensional vectors and coming up with a vector that's essentially a one-dimensional vector perpendicular to those two vectors, but if you start with 3-dimensional vectors, what the cross product is doing becomes clearer. And, yes, it was invented, but it was invented to deal with physical situations. It was pretty tough to invent, too. It's not a natural progression from 2-dimensional vectors (with complex numbers) to 3-dimensional vectors. A math system for 4-dimensional vectors (1 real component and three imaginary components) had to be developed and then set the real component to 0 to make the vector 3-dimensional instead of 4-dimensional.