B Confusion about the radius unit vector in spherical coordinates

Think about it. In three dimensions you need three quantities to define a point in space. They could be {x,y,z} or {r,θ,φ}. If I were to tell you that a fly is at position ##1.5~\hat r## meters from the corner of a room (the origin), where would you look to find it?

 

Because it makes the mathematical description in some cases easier to formulate and manipulate. Take the example of a point charge ##Q## at the origin. The electric field due to this charge can be written in spherical coordinated as$$\vec E(\vec r)=\frac{Q}{4 \pi \epsilon_0}\frac{\vec r}{r^3}.$$The same field in Cartesian coordinates is$$\vec E(x,y,z)=\frac{Q}{4 \pi \epsilon_0}\frac{x~\hat x + y~\hat y+z~\hat z}{(x^2+y^2+z^2)^{3/2}}.$$The top expression is easier to visualize as a radial field and often easier to work with if one has to take dot or cross products with other vectors or use vector calculus.

 

In the example of the electric field that I gave you, imagine a line from the origin where charge ##Q## is located to point ##P## where you want to find the field. The unit vector ##\hat r## is along this line and points from the origin to point ##P##. Its direction of course depends on where ##P## is. You can specify that direction in the Cartesian representation and write $$\hat r=\frac{x~\hat x + y~\hat y+z~\hat z}{(x^2+y^2+z^2)^{1/2}}.$$You tell me where point ##P## is, i.e. you give me ##x##, ##y##, and ##z## and I will be able to draw ##\hat r##. In terms of the standard spherical angles ##\theta## and ##\phi## that give an alternate way to find point ##P##, you have ##\hat r=\sin\theta \cos\phi~\hat x+\sin \theta \sin \phi~\hat y+\cos\theta ~\hat z##. You give me the angles and I will be able to draw ##\hat r##.

 

By definition you will never need any other basis vector in spherical coordinates when you express the position vector or any other vectors pointing radially. However, this is not the only possible vector. For example, your fly may be moving non-radially, which will give it a velocity vector with non-radial components. Same thing with the velocity vector of a planet orbiting a star. Since the orbit is typically almost circular, the radial component is small and the tangential component large.

 

Can you clarify what you mean? 'r' is not a vector. It is a real number indicating magnitude and has no direction. The vector with a direction is ##\vec r##, whose direction can be defined using a unit direction vector ##\vec u = \vec r / \mid \vec r \mid##. The unit vector direction ##\vec u ## can also be described using angles off of a coordinate system.