0

When I learn cross product, I find myself always using orthogonal basis vectors (e.g. $\hat{i}$, $\hat{j}$ and $\hat{k}$). But I am wondering does cross product depend on the orthogonality of the basis vectors? i.e. can I calculate the cross product of two vectors which are written in a set of non-orthogonal basis vectors? Why or why not?

Also, what if the basis vectors are changing (e.g. in cylindrical polar coordinates)? Why is cross product still defined in such a set of basis vectors?

How about for dot products? Does the dot product depend on the orthogonality or invariation of the basis vectors? Why or why not?

Bruce M
  • 245
  • In spherical polar coordinates we have $$ \hat{\boldsymbol{r}}\times\hat{\boldsymbol{\theta}}=\hat{\boldsymbol{\phi}},,\quad \hat{\boldsymbol{\theta}}\times\hat{\boldsymbol{\phi}}=\hat{\boldsymbol{r}},,\quad \hat{\boldsymbol{\phi}}\times\hat{\boldsymbol{r}}=\hat{\boldsymbol{\theta}} $$ – Kurt G. Jan 24 '24 at 19:09
  • and in cylindrical coordinates we have $$\hat{\boldsymbol{\rho}}\times\hat{\boldsymbol{\phi}}=\hat{\boldsymbol{z}},,\quad \hat{\boldsymbol{\rho}}\times\hat{\boldsymbol{z}}=-\hat{\boldsymbol{\phi}},,\quad \hat{\boldsymbol{\phi}}\times\hat{\boldsymbol{z}}=\hat{\boldsymbol{\rho}},$$ which should take us quite far. – Kurt G. Jan 24 '24 at 19:09

0 Answers0