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Relationships Among Unit Vectors
Recall that we could represent a point P in a particular system by just listing the 3
corresponding coordinates in triplet form:
x , y z , Cartesian
r , θ ,ϕ Spherical
and that we could convert the point P’s location from one coordinate system to
another using coordinate transformations.
Cartesian Æ Spherical Spherical Æ Cartesian
2 2 2
2 2
1
1
tan
tan
r x y z
x y
z
y
x
θ
ϕ
−
−
sin cos
sin sin
cos
x r
y r
z r
θ ϕ
θ ϕ
θ
Recall that we could represent a point P in a particular system using vectors:
x , y z , Cartesian
r , θ ,ϕ Spherical
or
P = a x + b y + c z Cartesian
P = a r + b θ + c φ Spherical
NOTE: The Cartesian system is taken to be the default coordinate system by
which all others are vector systems defined.
What happens if we want to convert vector information about a point P from one
coordinate system to another?
What we need are relationships or transformations between the various unit
vectors
Consider a point P in spherical coordinates with the vector form:
P = a r + b θ + c φ
Since
x , y , z for a orthogonal basis set as does
r , θ , φ , we can write
r , θ , φ in
terms of
x , y , z with the appropriate transformations of the form:
1 1 1
r = a x + b y + c z
2 2 2
θ = a x + b y + c z
3 3 3
φ = a x + b y + c z
To determine what coefficients a
i
, b
i
and c
i
are, we must take the dot product of
x ˆ , y ˆ , z ˆ with each of
r , θ , φ. To determine the value of the dot products, we can
use the following figure and make use of the geometry between spherical and
Cartesian coordinates:
If we wanted to write
r , θ , φ in terms of ˆ
x , y , z , we would need to use the angles
of θ and ϕ.
Ex.
1 1 1 1
x r ˆ ⋅ = a x ˆ + b y ˆ + c z ˆ= a
Note : a 1
is the projection of ˆ x onto ˆ r. To find a
1
requires a two step process:
Project ˆ x onto the line formed by ˆ r and its projection onto the xy plane
Project (1) onto ˆ r
The game can be played in reverse to find ˆ x , y ˆ , z ˆ in terms of
r ˆ , θ , φ ˆ :
x = sin θ cos ϕ r + cos θ cos ϕ θ −sinϕ φ
y = sin θ sin ϕ r + cos θ sin ϕ θ +cosθ φ
z = cos θ r −sinθ θ
APPLICATION:
One of the most common vectors we will deal with is the position vector, r. In
Cartesian form, it looks like this:
r = x x + y y + z z
This form of the vector is measuring the displacement from the origin to some
point P.
If the position vector is measuring the displacement from some starting location
other than the origin, say ( x o,
y
o,
z
o), we can represent this new starting coordinate
with the vector r o
relative to the origin. The new position vector would look like
this:
o o o
− = x − x + y − y + z − z
o
r r x y z
NOTE : We will come back and use this form at a later time. In the mean time, all
vectors will be measured with respect to the origin.
Now that we have different unit vectors for different systems and a way to go back
and forth between them, the position vector can be represented in various ways:
r = x x + y y + z z
r = r r
2 2 2
r = x + y + z r
r = r sin θ cos ϕ x + r sin θ sin ϕ y + r cosθ z
** Depending on the problem, select the vector form that will be the most useful
and the easiest to work with.
