What is the role of theta in representing polar vectors?

[Gzyousikai]

It is known that the vector in polar coordinate system can be expressed as $$\mathbf{r}=r\hat{r}$$. In this formula, we don't see $$\hat{\theta}$$ appear.
But after the derivation yielding speed, $$\mathbf{v}=\dot{r}\hat{r}+r\dot{\theta}\hat{\theta}$$. Where does theta come from? And how to define its magnitude and direction?

 

[grzz]

The $\dot{θ}$ appears because of the differentiation of the unit vector $\hat{\underline{r}}$.

 

[sophiecentaur]

It is known that the vector in polar coordinate system can be expressed as $$\mathbf{r}=r\hat{r}$$. In this formula, we don't see $$\hat{\theta}$$ appear.
But after the derivation yielding speed, $$\mathbf{v}=\dot{r}\hat{r}+r\dot{\theta}\hat{\theta}$$. Where does theta come from? And how to define its magnitude and direction?

The angle will turn up if the position vector is changing direction (if motion is not radial).
The direction of the unit theta vector will be normal to the unit r vector.

It can sometimes be difficult to see any point in using anything other than cartesian co ordinates - until you come upon a suitable problem, when it suddenly makes good sense.

 

[vanhees71]

You see everything most easily by expressing all vectors in cartesian coordinates. For polar coordinates you have
$$\vec{r}=r \cos \theta \vec{e}_1 + r \sin \theta \vec{e}_2.$$
Here, $\vec{e}_j$ are a Cartesian basis, i.e., two fixed orthonormalized vectors in the plane.
Then you get
$$\hat{r}=\cos \theta \vec{e}_1 + \sin \theta \vec{e}_2, \quad \hat{\theta}=-\sin \theta \vec{e}_1+\cos \theta \vec{e}_2.$$
The derivatives of the polar unit-basis vectors thus are
$$\partial_r \hat{r}=0, \quad \partial_{\theta} \hat{r}=\hat{\theta}, \quad \partial_r \hat{\theta}=0, \quad \partial_{\theta} \hat{\theta}=-\hat{r}.$$
From this you get
$$\partial_r \vec{r}=\hat{r}, \quad \partial_{\theta} \vec{r}=r \hat{\theta}$$
and thus, using the product and chain rule
$$\vec{v}=\frac{\mathrm{d}}{\mathrm{d}t} \vec{r}=\dot{r} \hat{r} + r \dot{\theta} \hat{\theta}.$$

 

[blue_raver22]

The polar coordinate system is a useful way to represent vectors, particularly in situations where the direction of the vector is important. In this system, vectors are expressed in terms of a magnitude (r) and a direction (\theta), rather than the traditional x and y components.

In the formula \mathbf{r}=r\hat{r}, the \hat{r} represents the unit vector in the radial direction, which is defined as the direction from the origin to a point on the vector. This unit vector does not depend on the angle \theta, as it is simply the direction of the vector from the origin.

In the second formula, \mathbf{v}=\dot{r}\hat{r}+r\dot{\theta}\hat{\theta}, both the magnitude and direction of the vector are changing with time. The first term, \dot{r}\hat{r}, represents the change in magnitude of the vector in the radial direction, while the second term, r\dot{\theta}\hat{\theta}, represents the change in direction of the vector. The \hat{\theta} represents the unit vector in the tangential direction, which is defined as the direction perpendicular to the radial direction.

The magnitude and direction of \theta can be defined in a number of ways, depending on the specific situation. For example, in a circular motion, \theta can be defined as the angle between the vector and the positive x-axis. In other situations, it may be more useful to define \theta as the angle between the vector and a specific reference direction.

In summary, \theta is an important component in polar vector representation, as it represents the direction of the vector. Its magnitude and direction can be defined in different ways depending on the specific context, but it is always necessary to consider it when dealing with polar vectors.