vr = sph2cartvec(vs,az,el) converts
the components of a vector or set of vectors, vs,
from their spherical basis representation to
their representation in a local Cartesian coordinate system. A spherical
basis representation is the set of components of a vector projected
into the right-handed spherical basis given by $$({\widehat{e}}_{az},{\widehat{e}}_{el},{\widehat{e}}_{R})$$.
The orientation of a spherical basis depends upon its location on
the sphere as determined by azimuth, az, and
elevation, el.

Start with a vector in a spherical basis located
at 45° azimuth, 45° elevation. The vector points along the
azimuth direction. Compute its components with respect to Cartesian
coordinates.

Vector in spherical basis representation specified as a 3-by-1
column vector or 3-by-N matrix. Each column of vs contains
the three components of a vector in the right-handed spherical basis $$({\widehat{e}}_{az},{\widehat{e}}_{el},{\widehat{e}}_{R})$$.

Azimuth angle specified as a scalar in the closed range [–180,180].
Angle units are in degrees. To define the azimuth angle of a point
on a sphere, construct a vector from the origin to the point. The
azimuth angle is the angle in the xy-plane from
the positive x-axis to the vector's orthogonal
projection into the xy-plane. As examples, zero
azimuth angle and zero elevation angle specify a point on the x-axis
while an azimuth angle of 90° and an elevation angle of zero
specify a point on the y-axis.

Elevation angle specified as a scalar in the closed range [–90,90].
Angle units are in degrees. To define the elevation of a point on
the sphere, construct a vector from the origin to the point. The elevation
angle is the angle from its orthogonal projection into the xy-plane
to the vector itself. As examples, zero elevation angle defines the
equator of the sphere and ±90° elevation define the north
and south poles, respectively.

Cartesian vector returned as a 3-by-1 column vector or 3-by-N
matrix having the same dimensions as vs. Each
column of vr contains the three components of
the vector in the right-handed x,y,z basis.

The spherical basis is a set of three mutually orthogonal unit
vectors $$({\widehat{e}}_{az},{\widehat{e}}_{el},{\widehat{e}}_{R})$$ defined
at a point on the sphere. The first unit vector points along lines
of azimuth at constant radius and elevation. The second points along
the lines of elevation at constant azimuth and radius. Both are tangent
to the surface of the sphere. The third unit vector points radially
outward.

The orientation of the basis changes from point to point on
the sphere but is independent of R so as you move
out along the radius, the basis orientation stays the same. The following
figure illustrates the orientation of the spherical basis vectors
as a function of azimuth and elevation:

For any point on the sphere specified by az and el,
the basis vectors are given by:

Any vector can be written in terms of components in this basis
as $$v={v}_{az}{\widehat{e}}_{az}+{v}_{el}{\widehat{e}}_{el}+{v}_{R}{\widehat{e}}_{R}$$.
The transformations between spherical basis components and Cartesian
components take the form