How to convert a right handed coordinate system to left handed? So I have a point p = (2,-5,1) and it is in a "right handed coordinate system" and I want to convert it into a left handed coordinate system. I tried googling it, but all the results have to do with Unity and I am not tying to program anything, I just want to understand the math behind it.
This is the exact question from the slides. 
"Given a point P = (2,-5,1) in a right handed coordinate system, what is the point is a left handed coordinate system?"
 A: By "left-handed" and "right-handed" I understand an orthonormal basis, i.e. unit length basis vectors that are all mutually (pair-wise) perpendicular.
Hold out your right hand and point your index finger at the screen, your middle finger (horizontally) to the left, and your thumb vertically. You have a right-handed coordinate system. 
Do the same with your left hand, and you'll notice that the middle fingers point in different directions. 
Hold your right hand and left hand out in front of you with the index fingers at the screen, the thumbs up, and the middle fingers pointing at each other. All fingers on the same hand at right angles to each other.
You should see that the difference between the right-handed and left-handed is that the middle fingers point in opposite directions.
Turn your hands towards yourself so you index fingers are tip-to-tip. Now the thumbs point in the same direction and the middle fingers point in the same direction. Only the index figures go against each other.
Go back to the start position, and rotate your hands so your thumbs point towards each other. Index finger and middle finger agree.
This is the difference between a left-handed and right-handed orthonormal coordinate system: one of the axes (fingers) is different. If $\{{\bf e}_1,{\bf e}_2,{\bf e}_3\}$ is right-handed then $\{-{\bf e}_1,{\bf e}_2,{\bf e}_3\},$
$\{{\bf e}_1,-{\bf e}_2,{\bf e}_3\}$, $\{{\bf e}_1,{\bf e}_2,-{\bf e}_3\}$ are all left-handed. Moreover, $\{-{\bf e}_1,-{\bf e}_2,-{\bf e}_3\}$ will also be left-handed. As Max correctly pointed out in the comments, an odd number of sign changed will change the chirality of the coordinate system, while an even number preserves it. 
Also $\{-{\bf e}_1,{\bf e}_2,{\bf e}_3\}$, $\{{\bf e}_1,-{\bf e}_2,{\bf e}_3\}$,  $\{{\bf e}_1,{\bf e}_2,-{\bf e}_3\}$ and $\{-{\bf e}_1,-{\bf e}_2,-{\bf e}_3\}$ are left-handed.
A: You can convert between right- and left-handedness by switching any two of the axes: for example, $(-5,2,1)$ would do the trick.
With that out of the way, I wonder about the quality of your slides. There is a spelling mistake in the question, and there is no context or explanation for how to do the problem. My advice would be to find a more reliable and complete reference elsewhere.
A: Simply inverting the $z$ coordinate will do the trick.
