Spherical Coordinates Unit Vectors

Unit vector is used to define direction of a vector, magnitude of unit vector is 1. Unit vector is indicated by lowercase alphabet having a cap on head. If we want to define 'i' as unit vector then it will be written as i^ and will be pronounced 'i – cap'.
Sometimes unit vector is also called as normalized vector. Lets assume that vector 'p' is given to us then its unit vector will be written as:
p^ = p-> / | p-> |
Where p -> is pronounced as p- vector and | p-> | is called magnitude of p- vector.
Unit vector can also be defined for Cartesian, spherical, cylindrical, and polar coordinate systems.
Now we will discuss Spherical Coordinates unit vectors.
Unit vectors used for spherical coordinate system are r ^, φ ^, and θ ^.
r ^ indicates direction in which radial distance increases.
θ ^ is the unit vector in the direction of increment of polar angle. It is usually taken as
π ≥ θ ≥  0.
φ ^  is the unit vector in direction of increment of azimuth angle.
spherical coordinates unit vectors in form of Cartesian coordinates are given as:
r ^ = (sin θ cos φ) x^+ (sin θ sinφ) y^+ (cos θ) z^,
θ ^ = (cos θ cos φ) x^+ (cos θ sin φ) y^ – sinθ z^,
φ ^ = (- sin φ) x ^+ (cos φ) y ^,
Unit vectors used in Cartesian coordinate system are i^, j^ and k^ and these unit vectors represent the direction along three axis x, y, and z- axis.
Similarly in cylindrical coordinate system, three unit vectors are ρ^, φ^, and z^.