The points in with , corresponding to the vectors of Euclidean norm equal to 1, form a circle with radius equal to 1 centered at the origin which we call the unit circle, see:
Each point on the unit circle can be written for some angle , which we refer to as the angle of direction or direction of the vector . This follows from the definition of and in Chapter Pythagoras and Euclid.
Any vector can be expressed as
where is the norm of , and is a vector of length one with the same direction as , and is the angle of direction of . This is the polar representation of . We call the direction of and the length of :
Vectors of length are given by where .
We see that if , where and , then has the same direction as . If then has the opposite direction. In both cases,the norms change with the factor ; we have .
If , where and , then we say that the vector is parallel to . Two parallel vectors have the same or opposite directions.
Example: We have