- How many vectors can there be in a given direction? Infinitely 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.

What is the direction of **v** = ( -3, 2, 4)^{T} ?