Why Is the Polar Form of the Position Vector Defined as r = r\vec{r}?

[sherlockjones]

If we have a position vector $$\vec{r} = x\vec{i} + y\vec{j}$$ why is the polar form of the equation $$r = r\vec{r}$$? Don't we need two parameters to define a postition? And $$r$$ is the magnitude of $$\vec{r}$$?

Thanks

 

[geoffjb]

If we have a position vector $$\vec{r} = x\vec{i} + y\vec{j}$$ why is the polar form of the equation $$r = r\vec{r}$$? Don't we need two parameters to define a postition? And $$r$$ is the magnitude of $$\vec{r}$$?

Thanks

The two parameters are $r$ and $\theta$.

http://mathworld.wolfram.com/PolarCoordinates.html

 

[kyle8921]

for your question. I can provide an explanation for the use of polar kinematics in describing the position of an object.

Firstly, it is important to understand that polar kinematics is a coordinate system that is used to describe the position and motion of an object in two-dimensional space. This system uses two parameters, a radial distance (r) and an angular displacement (θ), to define the position of an object.

In the given equation, the position vector \vec{r} is written in Cartesian coordinates, where x and y represent the horizontal and vertical components of the vector, respectively. However, in polar coordinates, the position vector is represented as r\vec{r}, where r is the magnitude of the vector and \vec{r} is the unit vector in the direction of the vector.

So why do we only need one parameter, r, to define the position in polar coordinates? This is because the magnitude of the vector, r, already includes information about both the distance and direction of the object from the origin. The angle θ, which is the other parameter in polar coordinates, is not needed as it is already taken into account in the unit vector \vec{r}.

In summary, the polar form of the equation r = r\vec{r} is a concise and efficient way to represent the position of an object in polar coordinates, where the magnitude of the vector, r, includes information about both the distance and direction of the object from the origin. I hope this explanation helps clarify the use of polar kinematics in describing the position of an object.