Cross product in cylindrical coordinates

[fishingspree2]

In my physics textbook we have
$d\vec{l}=\hat{z}dz$
and then it says
$d\vec{l}\times \hat{R}=\hat{\phi}\sin \left (\theta \right )dz$

How so? What is $\hat{z}\times\hat{R}$? If it is $\hat{\phi}$ then where does the sine come from?

 

[SteamKing]

Check the definition of cross product.

 

[fishingspree2]

Check the definition of cross product.

If i use the fact that a X b = |a| |b| sin(theta) then I understand where the sine comes from, it this case it would also mean that z X R is in the φ direction if $d\vec{l}\times \hat{R}=\hat{\phi}\sin \left (\theta \right )dz$ is correct.

but when I compute z X R using the 3x3 matrix
R φ z
0 0 1
1 0 0

I get +φ, and there is no sine.

 

[fishingspree2]

Any idea what am I missing?

 

[SteamKing]

You have a big advantage on anyone commenting here: You are pulling some equation from your (unnamed) physics textbook. How about showing us a little more information?

 

[Chestermiller]

It looks like $\hat{z}$ is a unit vector in the axial direction, $\hat{\phi}$ is a unit vector in the circumferential direction, $\hat{R}$ is a unit vector pointing from the origin in an arbitrary spatial direction, and $\theta$ is the angle between the unit vector $\hat{R}$ and the z axis.

$$\hat{R}=\sin(\theta)\hat{r}+\cos(\theta)\hat{z}$$

where $\hat{r}$ is a unit vector in the radial coordinate direction.