- How many vectors can there be in a given direction? Infinitly Many.
- How many UNIT vectors can there be in a given direction? One.

There can be infinitely many vectors in a given direction since the elements of
a vector are real numbers.
So, for example in two dimensions,
there are an infinite number of vectors
pointing along the x-axis.
The
column matrices for these vectors look like
(s, 0)^{T}.

The unit vector for a given vector
points in the same direction as that vector.
There is only one unit vector in a given direction.
There is only one value for the expression **v**/|**v**|.
There is only
one unit vector in the positive x direction,
represented by ( +1, 0)^{T}.

The same is true for other orientations and other number of dimensions. The picture shows vectors of various lengths but same orientation. Only one of them is a unit vector.

Recall the formula:

angle = arc tan( y/x )

This formula is not very useful in three dimensions. When there are three axes it is not enough to determine the angle between a vector and just one axis.

Often the *unit vector for a given vector*
is used to express the given vector's direction.
There is only one such unit vector,
so this description of direction is unique.
This works for all dimensions.