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Answer:

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

Unit Vectors as Directions

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.

QUESTION 6:

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