Solving Problem Involving Cross Product

[azizz]

Hey,

I have a problem that can be written in the following form:

u=v x w

where u, v, w are 3by1 vectors and x is the cross product.
now I want to write v in term of u and w, but I have no idea of how to get vector v out of the previous equation. Someone who can help me with this?

Thanks a lot

 

[lavinia]

Hey,

I have a problem that can be written in the following form:

u=v x w

where u, v, w are 3by1 vectors and x is the cross product.
now I want to write v in term of u and w, but I have no idea of how to get vector v out of the previous equation. Someone who can help me with this?

Thanks a lot

The cross product vector,u, points perpendicular to the plane spanned by w and v and its length is the area of the parallelogram that w and v span. Look at the simple case where w and v lie in the xy-plane. Is there a unique solution for v given w and u?

 

[azizz]

Sorry about that. Its more than obvious that it is impossible to solve the problem as I stated it before.

Let me give you some more information about my issue. And what I want to do. Perhaps that will make it more clear.

Suppose we have a wig with a centre of gravity x and a rotating point o.

-----------------------
| . . . . . . .x. . . . . . . .|
-----------0-----------
1. . . . . . . . . . . . . . . .2

(just ignore the points . . . . .)

Then we have three forces: Fg (gravity force working on x), F1 and F2 (forces to keep the wig in equilibrium, working on the points indicated by 1 and 2). The location of the cog x with respect to the rotating point o is denoted by r.
The moment due to the gravity working on the cog is computed as: M = r x F, with x being the cross product.
Now suppose I am able to measure M and F, and I want to compute the location of the cog, that is, I want to know r.
If I only do a measurement as indicated above, I can not determine r.
But if I now do two additional measurements by rotating the setup by +30 and -30 degrees I have three measurements:
M1 = R1 (r x F1)
M2 = R2 (r x F2)
M3 = R3 (r X F3)
where R is the euler rotation matrix to transform the coordinates in the rotated setup to the initial position as indicated in my figure. That means that for the first measurement R1=Identity.
So here comes my question again: is it now possible to compute r, if the only unknown variable in the three equations above is r?

Thanks again