Vectors: cross product question

[tony873004]

If a, b, and c are vectors, and a=b x c, and a and c are known, how do I solve for b?

b=a/c ? I don't think we've covered diving vectors, and since the cross-product is a special case of multiplying vectors (as opposed to dot product), I'm not sure this is allowed anyway.

I'm trying to compute dipole moment, given a torque and an electric field. In my above example, a is torque, and c is electric field. I need to solve for b.

Thanks!

 

[rocomath]

We know that a is orthogonal to vector b & c, hence the cross product. So, one way to know if two vectors are orthogonal is proving that the dot product is 0.

Let b=<b_1,b_2,b_3>

b\cdot(b \ x \ c)=0

There should be a property and perhaps an example on how to do this operation. If you are unable to find it, let me know and I will show you.

 

[tony873004]

Thanks for your reply.

There are similar examples in our notes where we find the torque, but not the dipole moment. At first glance, this looked like an easy problem. I'm a little confused by the direction you're going, since "a" has not been used.

I'm going to change the variables from a b & c to t (torque), p (dipole moment), and E (electric field) just to be consistent. (t is actually tau, but you get the point).

A needle suspended from a string hangs horizontally. The electric field at the needle’s location is horizontal with a magnitude 3.7*10³ N/C and is at an angle of 30° with the needle. There is no net electrical force acting on the needle, but the string exerts a torque of 3.7*10^-3 to hold the needle in equilibrium. What is the needle’s dipole moment?

so
t=3.7*10^-3
E= 3.7*10³ , 30°
solve for p

And from class notes,
t= p x E

 

[rocomath]

What are the points for vectors a & c? And I am using a, it's (b x c).

 

[tony873004]

sorry, I switched the variable names. a became t. b became p, and c became E.
So we bave t = p x E. Solve for p, with t and E given.