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[SOLVED] Basic vector question

dwsmith

Well-known member
Feb 1, 2012
1,673
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
Jan 26, 2012
4,193
Re: basic vector question

I don't think you can simplify further.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,780
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.