Confusion about the radius unit vector in spherical coordinates

In summary, spherical coordinates require three quantities (such as {x,y,z} or {r,θ,φ}) to define a point in three-dimensional space. The radius unit vector, or the unit vector along the line from the origin to a point, can be expressed in terms of the standard spherical angles θ and φ. This vector can be useful for mathematical descriptions and visualizing fields, such as the electric field of a point charge at the origin. However, in some cases, other vectors may be needed to fully describe a point or object, such as the velocity vector of a moving fly or orbiting planet. The radius, or distance from the origin, is not a vector but can be used to define a unit direction vector
  • #1
random_soldier
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If the radius unit vector is giving us some direction in spherical coordinates, why do we need the angle vectors or vice versa?
 
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  • #2
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?
 
  • #3
I mean I suppose in that case I would have to tilt my head by a certain θ and a certain Φ to see the fly but then my question is why the need to have r as a vector? I thought the whole point in spherical coordinates as opposed to cartesian was the fact that r was just the distance from the center and to actually arrive at the required point meant offsetting by a certain θ and a certain Φ.
 
  • #4
random_soldier said:
I mean I suppose in that case I would have to tilt my head by a certain θ and a certain Φ to see the fly but then my question is why the need to have r as a vector?
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.
 
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  • #5
Pardon my curtness but I think I am now just confused about what exactly the radius unit vector is giving the direction of. May I please know?
 
  • #6
random_soldier said:
Pardon my curtness but I think I am now just confused about what exactly the radius unit vector is giving the direction of. May I please know?
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##.
 
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  • #7
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.
 
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  • #8
random_soldier said:
I mean I suppose in that case I would have to tilt my head by a certain θ and a certain Φ to see the fly but then my question is why the need to have r as a vector?
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.
 

1. What is the radius unit vector in spherical coordinates?

The radius unit vector in spherical coordinates is a vector that points from the origin of a spherical coordinate system to a specific point on the surface of a sphere. It is represented by the symbol "r" and is typically written as r = (x, y, z).

2. How is the radius unit vector calculated in spherical coordinates?

The radius unit vector is calculated by using the spherical coordinate system, which includes the angles theta (θ) and phi (φ) and the distance r from the origin. The formula for calculating the radius unit vector is r = sin(θ)cos(φ)i + sin(θ)sin(φ)j + cos(θ)k, where i, j, and k are the unit vectors in the x, y, and z directions, respectively.

3. What are the components of the radius unit vector?

The components of the radius unit vector are the values of x, y, and z that represent the direction and length of the vector. These values are calculated using the angles theta (θ) and phi (φ) and the distance r from the origin.

4. How is the radius unit vector used in spherical coordinates?

The radius unit vector is used to determine the position of a point on the surface of a sphere in spherical coordinates. It is also used in various mathematical and scientific calculations involving spherical coordinates, such as finding the gradient, divergence, and curl of a vector field.

5. What is the difference between the radius unit vector and the position vector in spherical coordinates?

The radius unit vector and the position vector in spherical coordinates are both used to locate a point on the surface of a sphere. However, the radius unit vector represents the direction and length from the origin to the point, while the position vector represents the coordinates of the point with respect to the origin of the coordinate system.

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