# [SOLVED]Basic vector question

#### dwsmith

##### Well-known member
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

##### Indicium Physicus
Staff member
Re: basic vector question

I don't think you can simplify further.

#### Klaas van Aarsen

##### MHB Seeker
Staff member
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.