# Polar Representation of Vector

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

- ,

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