Vectors: cross product question

In summary, to solve for vector b when given vectors a and c, we can use the cross product formula a=b x c. In order to find the dipole moment, we need to solve for b by using the equation b=a/c. However, this may not be allowed since dividing vectors is not a common operation in this context. Another way to check if two vectors are orthogonal is by proving that their dot product is 0. In the given example of finding the dipole moment, we can use the formula t=p x E, where t represents torque, p represents dipole moment, and E represents the electric field. To solve for p, we need to rearrange the formula as p=t x E.
  • #1
tony873004
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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!
 
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  • #2
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 [tex]b=<b_1,b_2,b_3>[/tex]

[tex]b\cdot(b \ x \ c)=0[/tex]

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.
 
Last edited:
  • #3
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
 
  • #4
What are the points for vectors a & c? And I am using a, it's (b x c).
 
  • #5
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.
 

Related to Vectors: cross product question

1. What is a cross product?

A cross product is a mathematical operation that takes two vectors as input and produces a third vector as output. It is also known as a vector product or outer product.

2. How is a cross product calculated?

A cross product is calculated by taking the determinant of a 3x3 matrix composed of the unit vectors and the components of the two input vectors. The result is a vector that is perpendicular to both input vectors.

3. What is the geometric interpretation of a cross product?

The geometric interpretation of a cross product is that it produces a vector that is perpendicular to both input vectors, and its magnitude is equal to the area of the parallelogram formed by the two input vectors.

4. What are some applications of cross products?

Cross products are used in many fields, including physics, engineering, and computer graphics. They are used to calculate torque, magnetic fields, and 3D rotations.

5. Can a cross product be calculated for vectors in any dimension?

No, a cross product can only be calculated for vectors in three-dimensional space. In higher dimensions, a similar operation called the wedge product is used.

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