Magnitude and direction of a vector (-5,6)

Find the magnitude and direction of the vector $<-5,6>$

I found the magnitude:

$||v||=\sqrt{(-5)^2+6^2}=\sqrt{25+36}=\sqrt{61}$

In the direction this is what I did:

$\theta=\tan^{-1}(\frac{6}{-5})$

The inverse tangent gives me approximately -50.19°. When calculating vector direction when I get negative angles in degrees I just add to it $360°$ to find a co-terminal that is positive. I don't know if what I'm doing is right. I found a co-terminal that is positive adding $360°$ to $-50.19°$ which gives me $309.81°$.

I'm confirming my results with this website and the magnitude to see if my results are right but the direction says is $128.81°$

So I'm asking myself if what I'm doing when getting negative angles is right.

1 Answer

Remember that the tangent is negative in quadrants $2$ and $4$, so you shouldn't just routinely add $360$ degrees to your negative calculator value (that's basically assuming fourth quadrant (angle between $270$ and $360$ degrees) by default. Instead, figure out which quadrant the vector lies in. Note that $(-5,6)$ means negative $x$ coordinate and positive $y$ coordinate, and that means second quadrant (angle between $90$ and $180$ degrees). So the answer should be $180$ degrees minus the positive reference angle (what you get when you do $\arctan \frac 65$ [dropping the negative sign] on your calculator), which is equivalent to adding $180$ degrees to your negative calculator value (which you get when you evaluate $\arctan (-\frac 65) $ on most calculators).

The most proper approach to this sort of problem (finding the direction of a vector or the argument of a complex number) is always to ignore the sign of the ratio when evaluating the arctangent. That will give you the reference angle, which always lies in the first quadrant. Then you decide which quadrant the angle you want actually lies in based on the signs of $x$ and $y$. If first, do nothing more, accept the positive calculator value. If second, take $180$ degrees minus your positive calculator value. If third, take $180$ degrees plus your positive calculator value. If fourth, take $360$ degrees minus your positive calculator value.

If your ranges for the angles are defined differently (for instance with one commonly used convention for complex number arguments, the ranges often go from $-180 \ (-\pi) $ to $+180 \ (+\pi)$ rather than $0$ to $360 \ (2\pi)$ like in your case). In that scenario, you'll need to adjust your algorithm accordingly when working out the argument, but the principle is the same.