Vector Triple Product: Simplification Possible?

[Dustinsfl]

Given the basis $\{\mathbf{b},\mathbf{c},\mathbf{b}\times\mathbf{c}\}$.
We define the triple vector product as
$$
\mathbf{b}\times(\mathbf{b}\times\mathbf{c}) = (\mathbf{b}\cdot\mathbf{c})\mathbf{b} - b^2\mathbf{c}
$$
Can this be simplified further? We don't know if b and c are orthogonal just that they are linearly independent.

 

[Ackbach]

Re: basic vector question

I don't think you can simplify further.

 

[I like Serena]

Re: basic vector question

Given the basis $\{\mathbf{b},\mathbf{c},\mathbf{b}\times\mathbf{c}\}$.
We define the triple vector product as
$$
\mathbf{b}\times(\mathbf{b}\times\mathbf{c}) = (\mathbf{b}\cdot\mathbf{c})\mathbf{b} - b^2\mathbf{c}
$$
Can this be simplified further? We don't know if b and c are orthogonal just that they are linearly independent.

Nope.
Note that $\{\mathbf{b},\mathbf{b}\times\mathbf{c},\mathbf{b}\times(\mathbf{b}\times\mathbf{c})\}$ is an orthogonal basis.
Effectively you are looking at the Gram-Schmidt orthogonalization algorithm.