Why is the unit vector φ̂ perpendicular in polar coordinates?

[Istiak]

Homework Statement

Why angle direction is perpendicular?

Relevant Equations

vector

The unit vector r roof points in the direction of
increasing r with phi fixed; phi roof points in the direction of increasing phi
with r fixed. Unlike x roof, the vectors r roof and phi roof change as the position
vector r moves.
What I was thinking of the image is

Although, I was thinking why phi roof is perpendicular.

I was trying to understand that direction by the vector direction. I can't figure out that direction. Usually, I didn't do any vector of angle. I was wondering I didn't find any tutorial of angle direction in YT.

 

[PeroK]

What's the question exactly?

 

[vela]

Start with $\vec r = r(\cos \phi\,\hat i + \sin\phi\,\hat j)$ and find the direction of $\partial \vec r / \partial \phi$.

 

[Istiak]

Start with $\vec r = r(\cos \phi\,\hat i + \sin\phi\,\hat j)$ and find the direction of $\partial \vec r / \partial \phi$.

If I differentiate that then, I get

$$\frac{\partial \vec r }{\partial \phi}=r (\cos \phi \hat j - \sin \phi \hat i)$$

But, I was thinking what I should do with the equation.

 

[haruspex]

If I differentiate that then, I get

$$\frac{\partial \vec r }{\partial \phi}=r (\cos \phi \hat j - \sin \phi \hat i)$$

But, I was thinking what I should do with the equation.

What happens if you take the dot product of that with $\hat r$?

 

[PeroK]

If I differentiate that then, I get

$$\frac{\partial \vec r }{\partial \phi}=r (\cos \phi \hat j - \sin \phi \hat i)$$

But, I was thinking what I should do with the equation.

Show that vector is orthogonal to $\vec r$

 

[Istiak]

What happens if you take the dot product of that with $\hat r$?

$$\hat r \cdot \frac{\partial \vec r }{\partial \phi} = \hat r \frac{\partial \vec r }{\partial \phi} \cos \theta$$
$$=\hat r r (\cos \phi \hat j - \sin \phi \hat i) \cos \theta$$

 

[Istiak]

Show that vector is orthogonal to $\vec r$

How? :thinking Did you mean to graph?

 

[PeroK]

How? :thinking Did you mean to graph?

You could use the dot product - correctly, of course.

 

[PeroK]

$$\hat r \cdot \frac{\partial \vec r }{\partial \phi} = \hat r \frac{\partial \vec r }{\partial \phi} \cos \theta$$
$$=\hat r r (\cos \phi \hat j - \sin \phi \hat i) \cos \theta$$

That's an original approach to say the least!

 

[Istiak]

You could use the dot product - correctly, of course.

To me $$\hat r r$$ represents the direction of phi is toward r. But, I think $$(\cos \phi \hat j - \sin \phi \hat i) \cos \theta$$ this are representing direction of phi is perpendicular. But, I am saying that by looking at original picture. Without that, I can't say that. So, how can I see that that's really perpendicular.

 

[PeroK]

To me $$\hat r r$$ represents the direction of phi is toward r. But, I think $$(\cos \phi \hat j - \sin \phi \hat i) \cos \theta$$ this are representing direction of phi is perpendicular. But, I am saying that by looking at original picture. Without that, I can't say that. So, how can I see that that's really perpendicular.

Note that it's either $\phi$ or $\theta$, not both. Also, look up the dot product in Cartesian coordinates and note that:

Start with $\vec r = r(\cos \phi\,\hat i + \sin\phi\,\hat j)$

And

$$\frac{\partial \vec r }{\partial \phi}=r (\cos \phi \hat j - \sin \phi \hat i)$$