Cross product direction in Spherical coordiantes

Question

The cross product of a vector $\vec{m}$ with $\hat{r}$ gives $m\sin\theta \hat{\phi}$ as claimed by the author.

This would probably be trivial but a lot of these subtle technicalities were not encountered in my first year multivariate course. This is slowing down my progression considerably in Physics.

I want to know how $\hat{\phi}$ is determined.

Edit:

I'd like to add that the direction due to the above cross product must be perpendicular to both $\vec{m}$ and $\hat{r}$. Clearly, It must lie on the xy-plane for it to be mutually perpendicular to $\vec{m}$ and $\hat{r}$. But the $\phi$ is elusive.

Hint: use a $\LaTeX$ symbol \sin for sine function. – CiaPan Sep 01 '16 at 07:51 — CiaPan, Sep 01 '16 at 07:51

score 1 · Accepted Answer · answered Sep 01 '16 at 10:22

1

Let the unit vectors in spherical coordinates be $\hat{\rho},\hat{\theta},\hat{\phi}$.

Note that $\vec{m}=m \hat{z}$, that $\hat{z}=\cos\theta \hat{\rho}-\sin\theta \hat{\theta}$ and that $\hat{\theta}\times\hat{\rho}=-\hat{\phi}$.

So the cross-product is $$\vec{m}\times \hat{\rho}=m(\cos\theta \hat{\rho}-\sin\theta \hat{\theta})\times\hat{\rho}=m\sin\theta \hat{\phi}.$$

answered Sep 01 '16 at 10:22

Bobson Dugnutt

10,672

1

@Mathematicing Did this answer your question? Otherwise, please do let me know. – Bobson Dugnutt Sep 02 '16 at 20:44
Can you give me a detailed cross product workings? – Mathematicing Sep 04 '16 at 09:28
Is $\rho$=1? or is it a linear combination of some basis? – Mathematicing Sep 04 '16 at 09:30
Yes, the length $\rho$ is 1, as it is a unit vector. What part of the cross-product workings is causing you trouble? Remember that cross-products are distributive, i.e., if $a,b,c$ are 3D vectors, then $a\times(b+c)=a\times b+a \times c.$ – Bobson Dugnutt Sep 04 '16 at 09:55
@Mathematicing Also note that $\rho$ is a basis vector. – Bobson Dugnutt Sep 04 '16 at 10:10
Actually, I got it. I was over thinking it. Thanks for the assistance. – Mathematicing Sep 04 '16 at 10:17
@Mathematicing No problem :) – Bobson Dugnutt Sep 04 '16 at 10:54

score 1 · Answer 2 · answered Sep 01 '16 at 11:06

Compare the spherical coordinates and Cartesian coordinates: $$\vec{n}(r, \varphi, \theta) = \begin{cases} x = r \cos(\varphi) \sin(\theta) \\ y = r \sin(\varphi) \sin(\theta) \\ z = r \cos(\theta) \end{cases}$$

By definition, the spherical coordinate system unit vectors are $$\hat{r} = \begin{cases} x = \cos(\varphi) \sin(\theta) \\ y = \sin(\varphi) \sin(\theta) \\ z = \cos(\theta) \end{cases}, \; \hat{\varphi} = \begin{cases} x = -\sin(\varphi) \\ y = \cos(\varphi) \\ z = 0 \end{cases}, \; \hat{\theta} = \begin{cases} x = \cos(\varphi) \cos(\theta) \\ y = \sin(\varphi) \cos(\theta) \\ z = -\sin(\theta) \end{cases}$$

If $$\vec{m} = \begin{cases} x = 0 \\ y = 0 \\ z = m \end{cases}$$ then $\vec{m} \times \hat{r}$ in Cartesian coordinates is $$\vec{m} \times \hat{r} = \begin{cases} x = -m \sin(\varphi) \sin(\theta) \\ y = m \cos(\varphi) \sin(\theta) \\ z = 0 \end{cases}$$ and $m \sin(\theta) \hat{\varphi}$ is $$m \sin(\theta) \hat{\varphi} = \begin{cases} x = -m \sin(\theta) \sin(\varphi) \\ y = m \sin(\theta) \cos(\varphi) \\ z = 0 \end{cases}$$ The two are obviously equal.

Cross product direction in Spherical coordiantes

2 Answers2