Cross-products and the right-hand rule

[Kavorka]

I understand how to do problems relating to this, I just don't really understand the concept behind it.

Why is the torque vector in rotational motion directed out of the plane of rotation? I can't wrap my head around why it would be directed that way in a real-world set of mind.

What does a cross-product actually represent? I understand that its magnitude is the area of a parallelogram formed by the two crossed vectors. I'm not sure why the determinant of a 3x3 matrix actually means something however. I know what it means in terms of mathematics, but not in terms of real-world forces and physics that I can conceptualize.

 

[paisiello2]

A torque represents a rotation about an axis. So the vector is just pointing in the direction of the axis.

A cross product is the amount of torque about an axis that two vectors cause where the axis can be in any direction. It can also be considered a measurement of the amount of perpendicularity between two vectors while the dot product is the measurement of the amount of parallelism.

A determinant of a 3x3 matrix generally can be thought of as the volume that 3 vectors form. So if one of the vectors is the unit vector then the volume becomes the area of a parallelogram.

 

[Kavorka]

Hm that makes more sense. Is the choice of the vector pointing in or out when the force is applied clockwise or counterclockwise just arbitrary?

 

[paisiello2]

Completely.

 

[gneill]

Hm that makes more sense. Is the choice of the vector pointing in or out when the force is applied clockwise or counterclockwise just arbitrary?

It depends upon your choice of coordinate system. Cartesian coordinate systems can be right-handed (the usual case) or left-handed. In a right-handed coordinate system you use the right-hand rule to find the resultant direction of a cross product, and $\hat{i} \times \hat{j} = \hat{k}$. In a left-handed coordinate system $\hat{i} \times \hat{j} = -\hat{k}$ and you'd use the left-hand rule to find the direction.

For a given coordinate system the direction of the resultant of a cross product of two vectors is not arbitrary. If it were arbitrary, conservation laws like conservation of angular momentum would not hold because the addition of angular momentum vectors would then be arbitrary, too, given arbitrary choices for their directions.

 

[paisiello2]

So is the OP right-handed or left-handed?

 

[Kavorka]

right-handed of course!