Polar Representation of Vector

The points a=(a_1,a_2) in R^2 with \vert a\vert =1, corresponding to the vectors a 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 a on the unit circle can be written a=(\cos(\theta ),\sin(\theta )) for some angle \theta, which we refer to as the angle of direction or direction of the vector a. This follows from the definition of \cos(\theta ) and \sin(\theta ) in Chapter Pythagoras and Euclid.

Any vector a=(a_1,a_2)\neq(0,0) can be expressed as

  • a=\vert a\vert \hat a=r (\cos(\theta ),\sin(\theta )),

where r=\vert a\vert is the norm of a, and \hat a=(a_1/\vert a\vert,a_2/\vert a\vert ) is a vector of length one with the same direction as a, and \theta is the angle of direction of \hat a. This is the polar representation of a. We call \theta the direction of a and r the length of a:

Vectors a of length r are given by a=r(\cos(\theta),\sin(\theta))=(r\cos(\theta),r\sin(\theta)) where r=\vert a\vert.

We see that if b=\lambda a, where \lambda >0 and a\neq 0, then b has the same direction as a. If \lambda <0 then b has the opposite direction. In both cases,the norms change with the factor \vert\lambda\vert; we have \vert b\vert =\vert\lambda\vert\vert a\vert.

If b=\lambda a, where \lambda\neq 0 and a\neq 0, then we say that the vector b is parallel to a. Two parallel vectors have the same or opposite directions.

Example: We have

  • (1,1)=\sqrt{2}(\cos(45^\circ ),\sin(45^\circ )),
  • (-1,1)=\sqrt{2}(\cos(135^\circ ),\sin(135^\circ )).

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