Simplifying a dot and cross product expression

Question

I am trying to solve or simplify $$ \left[ \,f\cdot \frac{(u\times v)}{||u\times v||_2} \right]\frac{(u\times v)}{||u\times v||_2} = 0 $$ where $f$ is a unit vector and $u,v$ are vectors.

Is there a way to simplify this?

I have been trying to use some of the triple product identities e.g. ${\displaystyle (\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ))\,\mathbf {a} =(\mathbf {a} \times \mathbf {b} )\times (\mathbf {a} \times \mathbf {c} )}$, but have only really made things more complicated.

Answer 1

$u,v$ cannot be parallel, otherwise the LHS would be undefined. Then your equation simplifies to

$$f\cdot(u\times v)=0$$ which expresses that the vectors $f,u,v$ are linearly dependent.

$$\lambda f+\mu u+\nu v=0,$$ with $\lambda,\mu,\nu$ not all zero.

Answer 2

Because $0$ on the right side means zero vector and $u \times v $ is non-zero vector then it must be $$ \,f\cdot (u\times v) = 0 $$
($0$ here is a scalar) .

Further I think it can't be simplified but it can be written alternatively in the matrix form $f^TS(u)v=0$ where skew-symmetric matrix $S(u)=[ u \times i \ \ \ \ u \times j \ \ \ \ u \times k ]$ is generated from the vector products of $u$ with standard basis vectors $i,j,k$.