Convert unit vector from cartesian to spherical coordination

In summary, unit vectors in spherical coordinates represent the direction and magnitude of a vector in a three-dimensional space, defined by the angles θ and φ. To convert a vector from Cartesian to spherical coordinates, use the formula r = √(x² + y² + z²), θ = cos⁻¹(z/r), φ = tan⁻¹(y/x). If the vector is not of unit length, divide each component by its magnitude, r. A unit vector is different from a position vector, which represents the coordinates of a point in space. An example of converting a unit vector from Cartesian to spherical coordinates is given using the vector (1,1,1).
  • #1
assassin2811
1
0
i have a problem :
A small loop antenna in free space and centered about the origin on the xy-plane is producing a
(far-field) radiation electric field (in phasor notation) :
http://postimg.org/image/63tm76h5l/

and their solution :
http://postimg.org/image/6mdm6roh9/

i don't understand how can they replace (-x^) with (∅^.sinθ), can anyone show me how to convert unit vectors in cartesian coordination (x^, y^, z^) into spherical (r^, θ^, ∅^), thanks
 
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Related to Convert unit vector from cartesian to spherical coordination

1. What is the formula for converting a unit vector from Cartesian to spherical coordinates?

The formula for converting a unit vector from Cartesian to spherical coordinates is:

r = √(x² + y² + z²), θ = cos⁻¹(z/r), φ = tan⁻¹(y/x)

2. Can you explain the concept of unit vectors in spherical coordinates?

Unit vectors in spherical coordinates represent the direction and magnitude of a vector in a three-dimensional space. They are defined by the angles θ and φ, which represent the inclination and azimuth, respectively, and have a magnitude of 1.

3. How do you convert a unit vector from Cartesian coordinates if the vector is not of unit length?

To convert a vector from Cartesian coordinates to spherical coordinates if the vector is not of unit length, you need to divide each component of the vector (x, y, z) by its magnitude, r. This will give you the unit vector in spherical coordinates.

4. What is the difference between a unit vector and a position vector in spherical coordinates?

A unit vector in spherical coordinates represents the direction and magnitude of a vector, while a position vector represents the coordinates of a point in space. The position vector is defined by the radius, inclination, and azimuth, while the unit vector is defined by the angles θ and φ.

5. Can you provide an example of converting a unit vector from Cartesian to spherical coordinates?

As an example, let's convert the unit vector (1,1,1) from Cartesian to spherical coordinates. First, we need to calculate the magnitude, r, which is √(1² + 1² + 1²) = √3. Then, we can find the angles θ and φ using the formulae: θ = cos⁻¹(1/√3) ≈ 54.74° and φ = tan⁻¹(1/1) = 45°. Therefore, the unit vector in spherical coordinates is (√3, 54.74°, 45°).

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