Understanding the Direction of 'r' Vector in Torque and Biot-Savart Law

[hansen4332]

So I understand the with torque, T= r x F: r is a vector that points from axis to point of force, correct?

I am confused about the direction of r is a couple of other circumstances, and I wanted to know if there is handy rule.

For example a linear momentum, this diagram, does 'r' also point from center of orbit to the mass?

Also, in Biot-Savart law:

$$d{\bf{B}} = \frac{{\mu _0 }}{{4\pi }}\frac{{Id\ell \times {\bf{\hat r}}}}{{r^2 }}$$

what is the direction of 'r' ... from current to the point or the other way around??

Is there a handy rule?

Thanks a lot!

 

[A.T.]

Is there a handy rule?

For all possible meanings of "r"? No, but usually it's a vector describing a position, relative to some reference point.

 

[andrewkirk]

You are correct in all three cases. For angular momentum the r is from axis to CoM of the object whose angular momentum is being measured. For Biot-savart it is the vector from the wire to the point of measurement.

The handy rule is that a formula used to define a physical quantity in terms of a cross product will be written in an order that makes the vector of the physical quantity point the way we want it to point. Note how the r is the first argument to the cross product for torque and ang mom and the second argument for Biot-Savart. That's because that's where it needs to be to make the formula work.

 

[David Lewis]

...with torque, T= r x F: r is a vector that points from axis to point of force, correct?

You've got the sense of r correct but I wouldn't use the term point of force . You could say instead the head of displacement vector r is a point on the force vector's line of application.

 

[David Lewis]

Correction: Should be line of action, not line of application.