Can the Inverse of an Inner Product Be Described as a Function of Two Vectors?

[steem84]

my question is:

if a.b=c

with
a=any vector
b=any vector
.=inner product
c=resulting scalar

is there a way to describe a=f(b,c)?

Thanks

 

[Fredrik]

No. a isn't completely determined by b and c, since you can replace a with a+d in your first equation, if d is orthogonal to b.

 

[D H]

No. Think of it this way: Suppose you find vectors a and d such that a.b = c and d.b=0 (i.e., d is orthogonal to b). Then for any scalar k, (a+k*d).b=c also. In other words, the solution to a.b=c is not unique.

 

[steem84]

Ok thanks!

but it did not solve my problem regarding vortex dynamics...

I would like to use the the law of Biot and Savart to determine the velocity field induced by a vortex filament. I am trying to do this by rewriting the equation of the circulation to a function for the vorticity.

To do this, I am using the Dirac Delta function for another integral for the circulation. This way i can equal the two integrands. But from that expression I would like to write the vorticity as a function of the rest (to substitute it in Biot ans Savart). But from your explanations I see that this is not possible??

 

[steem84]

already solved it...

w.n = the magnitude of the vector=length(w)
x=f(s)
dx/ds / length(dx/ds) = unity direction of the vector

so the vector is w is determined to be (w.n)*(dx/ds / length(dx/ds))