# 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

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