# How do we really get the angle of a vector from the components?

Usually when people discuss getting the polar form of a vector $$v$$, they present the following two formulas:

$$\text{Magnitude}(v) = \sqrt{x^2 + y^2}$$

$$\text{Angle}(v) = \arctan \left(\frac{y}{x} \right)$$

$$\text{ Where } \space v = \begin{bmatrix} x \\ y \end{bmatrix}$$

I believe that this formula for the angle is only partially true. I think a better and more complete formula for the angle should be:

$$\text{Angle}(v) = \begin{cases} \arctan \left(\frac{y}{x} \right) &; \space x \gt 0 \\ \pi +\arctan \left(\frac{y}{x} \right) &; \space x \lt 0 \\ {\begin{cases} \operatorname{sign}(y) \frac{\pi}{2} &; y \neq 0 \\ \text{undefined} &; \space y = 0 \end{cases}} &; x = 0 \end{cases}$$

Is there some sort of way to simplify this or to better express this, or is this it?

• The "atan2" function was devised to help with this. – Blue Jan 20 at 14:59
• BTW: You can see atan2 being used in the "Converting between polar and Cartesian coordinates" section of Wikipedia's "polar coordinate system" entry. Who are these people you've been talking-to who "usually" just use arctan? – Blue Jan 20 at 15:06
• That's unfortunate. :/ – Blue Jan 20 at 15:13
• @Blue Thank you for your sympathy. :) – Gustav Jan 20 at 15:16
• I'll note that the computer algebra Mathematica uses a "smart" ArcTan function: when fed just one argument, it gives the regular inverse-tangent; when given both $x$ and $y$ arguments, it gives the quadrant-savvy version of the angle (equivalent to atan2). – Blue Jan 20 at 15:20

If you want an answer in form of a mathematical function definition, using only functions that were in common use in undergraduate instruction fifty years ago, I do not think you can do much better than the excellent answer by Rhys Hughes. As noted in the answer, this is how mathematicians often define a function equivalent to yours in textbooks. The only detail you might want to add is something to deal with the case $$x = y = 0.$$

Note that the formulas in that answer do not tell you how to find $$\theta,$$ but they do uniquely identify the output of the function for any possible input. There is always some value of $$\theta$$ that will satisfy both equations when $$x^2 + y^2 \neq 0,$$ and there will never be more than one value of $$\theta$$ that satisfies both equations.

If you want a formula to compute the angle using only functions that were in common use in undergraduate instruction fifty years ago, I think the formula you wrote is close to the best you can get, though I would handle one or two cases a bit differently.

If you want a nice way to represent your function in other formulas, you can borrow the two-parameter arc tangent function that is defined in many software packages. That is, define a function $$\operatorname{atan2}(y, x)$$ whose value is the angle of the vector $$\begin{bmatrix} x \\ y \end{bmatrix}.$$ You can define $$\operatorname{atan2}(y, x)$$ either in the style of the complex analysis textbooks described in the other answer, or you can define it in your style:

$$\operatorname{atan2}(y, x) = \begin{cases} \arctan\left(\frac yx\right) & x \gt 0 \\ \arctan\left(\frac yx\right) + \pi \quad & x \lt 0, \ y \geq 0 \\ \arctan\left(\frac yx\right) - \pi \quad & x \lt 0, \ y < 0 \\ \frac\pi2 & x = 0,\ y > 0 \\ -\frac\pi2 & x = 0,\ y < 0 \\ \text{undefined} & x = 0,\ y = 0. \end{cases}$$

When this is implemented in software I think the "undefined" case usually returns $$0.$$

If you have to write formulas involving the direction angles of several two-dimensional vectors in terms of their components, then you might find the notation $$\operatorname{atan2}(y, x)$$ convenient.

• @Gustav Wow! I mean wow squared. You complain that "they" leave a little bit for you to figure out instead of giving you exactly what you want on a silver platter, then you speculate this is because they are lazy? Wow. (This is an example of what they mean by "irony". If you visit the_internet.com you can apply for a refund of the monry you paid...) – David C. Ullrich Jan 20 at 16:29
• @Gustav Not frustrated at all, more amused. – David C. Ullrich Jan 20 at 16:36
• @Gustav I understood the question. And I have no problem with anything you said about the answer - my comment had to do with you complaining about him being lazy. (Do you know what the word "irony" means?) – David C. Ullrich Jan 20 at 16:45
• No problem; I've deleted earlier comments. – David K Jan 20 at 17:12
• @David C. Ullrich. Could you also remove your previous comments as they no longer serve any purpose? – Gustav Jan 20 at 17:20

This looks very similar to complex numbers, where $$\text{Magnitude}(v)$$ is displayed by $$|v|$$ and $$\text{Angle}(v)$$ is written $$\text{arg}(v)$$. We have that $$\arg(v)$$ is the unique angle $$\in (-\pi, \pi]$$ where:

$$\cos\theta =\frac{x}{|v|}=\frac{x}{\sqrt{x^2+y^2}}$$ and $$\sin\theta=\frac{y}{|v|}=\frac{y}{\sqrt{x^2+y^2}}$$

• When working with only arctan, if we look at its graph, we can clearly see that it only gives values in the interval [$\frac{-\pi}{2}$ , $\frac{\pi}{2}$]. This only covers the first and fourth quadrants. So, in order to access the second and third quadrants, we need another definition. On top of that we need a definition for when the denomination in the usual definition would be zero, etc. – Gustav Jan 20 at 15:22
• When the denominator equals to zero, the 2-norm or magnitude of the vector is 0 anyway. In other words, the 0 vector has no angle (Linear Algebra tells us that 0 is in the null space of any system fully described by the $R^2$). – Joel Biffin Jan 20 at 15:52
• @Joel When the denominator is zero, only one of the components is zero, not both. So no, the magnitude is not zero. – Gustav Jan 20 at 15:58
• @Gustav you literally called $\sqrt{x^2+y^2}$ the magnitude in your question! – Rhys Hughes Jan 20 at 16:01
• 2 comments back, I accidentally wrote denomination instead of denominator. With this I mean the value of x. Sorry about that. – Gustav Jan 20 at 16:03