1.3 Polar form in the Euclidean plane

In this section we will focus on vectors in $\mathbb{R}^2$, and consider a different way of representing them. As well as defining a vector in $\mathbb{R}^2$ based on its components in the $x$ and $y$ directions, we could also define it based on its length and its angle from the $x$-axis. This is known as polar form, and is illustrated in Figure 1.4.

$An arrow from the origin representing vector $v$ is pictured on the Euclidean plane. Dashed lines go from the point of the arrow to the two axes, with the point where the vertical axis meets the horizontal dashed line labelled $y$ and the point where the horizontal axis meets the vertical dashed line labelled $x$. A curved line goes from the horizontal axis to the vector representing the angle that the vector makes witht the axis. This angle is labelled $\theta$. The arrow representing the vector is also labelled by $\lambda$ representing the length of the vector$

Figure 1.4: A vector $v$ in $\mathbb{R}^2$ represented by Cartesian coordinates $(x,y)$ or by polar coordinates $\lambda, \theta$. We have $x=\lambda\cos\theta$, $y=\lambda \sin\theta$, $\lambda=\sqrt{x^2+y^2}$ and $\tan\theta=\dfrac{y}{x}$.

In particular, a unit vector has length one, hence all unit vectors lie on the circle of radius one in $\mathbb{R}^2$, and a unit vector is determined solely by its angle $\theta$ with the $x$-axis. By elementary geometry we find that the unit vector with angle $\theta$ to the $x$-axis is given by \[\label{eq:unitvec} u(\theta):=\begin{pmatrix}\cos\theta\\ \sin\theta\end{pmatrix}.\]

We can then multiply by a scalar order to obtain any vector in $\mathbb{R}^2$, and this gives us a unique vector. In particular, the scalar that we multiply our unit vector by is the norm of the vector.

Theorem 1.18:

For every $v\in\mathbb{R}^2$, $v\neq 0$, there exist unique $\theta\in [0,2\pi)$ and $\lambda\in (0,\infty)$ with \[v=\lambda u(\theta)\]

Proof.

Given $v=\begin{pmatrix}v_1\\ v_2\end{pmatrix}\neq 0$ we have to find $\lambda>0$ and $\theta\in [0,2\pi)$ such that \[\begin{pmatrix}v_1\\ v_2\end{pmatrix}=\lambda u(\theta)=\begin{pmatrix}\lambda \cos \theta\\ \lambda \sin\theta\end{pmatrix}.\] Since $\lVert\lambda u(\theta)\rVert=\lambda\lVert u(\theta)\rVert=\lambda$ (note that $\lambda>0$, hence $\lvert\lambda\rvert=\lambda$) we get immediately \[\lambda=\lVert v\rVert.\] To determine $\theta$ we have to solve the two equations \[\cos\theta=\frac{v_1}{\lVert v\rVert} ,\quad \sin\theta=\frac{v_2}{\lVert v\rVert} ,\] which is in principle easy, but we have to be a bit careful with the signs of $v_1,v_2$. If $v_2>0$ we can divide the first by the second equation and obtain $\cos\theta/\sin\theta=v_1/v_2$, hence $\theta=\cot^{-1} \frac{v_1}{v_2},$ that is \[\theta=\arctan\frac{v_2}{v_1}\in (0,\pi) .\] If $v_1>0$ and $v_2 < 0$ we have $\arctan (v_2/v_1) \in (-\frac{\pi}{2},0)$ and so $\theta=2\pi+\arctan (v_2/v_1)$; analogous arguments apply in the remaining cases and this is illustrated in Figure 1.5.

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$The Euclidean plane is pictured, with a pair of axes dividing it into quadrants. The first (top-right) quadrant reads $v_1, v_2 > 0, 0< \theta <\frac{\pi}{2} \theta=\arctan\left(\frac{v_2}{v_1}\right)$. The second (top-left) quadrant reads $v_1<0, v_2 > 0, \frac{\pi}{2}< \theta < \pi, \theta=\pi+\arctan\left(\frac{v_2}{v_1}\right)$. The third (bottom-left) quadrant reads $v_1, v_2 < 0, \pi< \theta < \frac{3\pi}{2}, \theta=\pi+\arctan\left(\frac{v_2}{v_1}\right)$. The fourth (bottom-right) quadrant reads $v_1>0, v_2 < 0, \frac{3\pi}{2}< \theta < 2\pi, \theta=2\pi+\arctan\left(\frac{v_2}{v_1}\right)$.$

Figure 1.5: How to calculate the argument for $(v_1,v_2)$ in each quadrant of the Euclidean plane.

The converse of this result also holds, that is given $\theta\in [0,2\pi)$ and $\lambda\geq 0$ we get a unique vector with direction $\theta$ and length $\lambda$: \[v=\lambda u(\theta)=\begin{pmatrix}\lambda\cos\theta\\ \lambda\sin\theta\end{pmatrix}.\]

Exercise 1.19:

Let $v=\begin{pmatrix} -1 \\ \sqrt{3}\end{pmatrix}$. Find $\lambda\in (0, \infty]$ and $\theta \in [0,2\pi)$ such that $v=\lambda u(\theta)$.

There are many practical situations where the polar form of a vector might be more useful than the Cartesian form, for example a ship navigating from a port may travel a certain distance at a given angle. In the next chapter, we will look at an extended example of vectors in $\mathbb{R}^2$, namely the complex plane, see how complex numbers can be viewed as vectors and how their polar form provides a useful insight into the geometric interpretation of multiplication.