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.