I have been searching about the cross product and still can't grasp the physical intuition of it. As far as I know, mathematically the cross product is a tool that creates a new vector perpendicular to the two given independent vectors. This is useful for describing a plane with a single vector instead of two. Some definitions include that the magnitude of the cross vector is the area of the parallelogram defined by the two vectors of which I don't see the relation or how it makes sense since it's just a result and not how it was derived, hence why is there a sine in that formula.
But the main thing I don't get is, since the cross is a vector as well then in physics it has a magnitude and direction, so it should describe a phenomenon. What phenomenon is it? And then most of its uses or application are in circular motions (angular momentum, torque, etc), what about linear motions? Wouldn't a cross product of two vectors in linear motion give us a perpendicular vector as well? What is that vector representing? What would be the cross product of two force vectors?
Best Answer
The cross product of two vectors is really a bivector. It has a magnitude and a direction, but the magnitude is an area instead of a length, and the direction is a plane instead of a line.
Like two vectors can point in opposite directions while lying on the same line, two bivectors can "point" in opposite directions while lying in the same plane. You can think of the directions as clockwise and counterclockwise, though which of those is which depends on which side of the plane you're on.
Bivectors are useful for things that lie in a plane and have a clockwise/counterclockwise direction and a magnitude, like angular velocity.
In three dimensions (and only in three dimensions), you can identify a bivector with a vector perpendicular to the plane of the bivector, whose length is the bivector's area. Because of this, bivectors are usually not taught as such. Instead, you have a cross product that produces another vector, whose direction is given by the right hand rule.