How to calculate the angles that define the position your arm is in

[CraigH]

I have 3 points $S$, $E$, and $W$ each with $x$, $y$, and $z$ positions.

$S$ is always at $[1,1,1]$

$S$ is connected to $E$, and $E$ is connected to $W$ by a line.

I want to calculate the azimuth and polar angles between $E$ and the vertical, and the azimuth and polar angles between $E$ and $W$, using the coordiantes I have for all of them. How would I do this?

 

[Simon Bridge]

construct vectors for the lines, convert to spherical, polar coordinates.
note. E us a point, it cannot have an angle to anything.

 

[CraigH]

Thanks for your reply :)

note. E us a point, it cannot have an angle to anything.

E is a point however it will have angles between itself and the vertical

 

[Simon Bridge]

No - a point cannot have angles between itself and anything else.
You can see this easily by drawing a point, and drawing a "vertical", but do not draw in an origin.
What is the angle between the point and the line you just drew? Doesn't make sense does it?

The vector pointing from the origin to E will have angles between itself and the vertical, as will any line passing through the origin and E. However, any line passing through E, any vector pointing from any point to E, will have an angle to the vertical, so you have to specify. Just saying "the angle of E" is too vague. We may be able to infer something from the context but, in maths at least, it is often a good idea to make implicit things explicit: it helps you cut through common confusions.

The way to tackle your problem is to construct the lines you want the angles between and use geometry.
One approach is the shift the axes so the z-axis lies along one line, and the origin is at the corner, then represent the point you are interested in with spherical coordinates. The azimuthal angle that results will also be the angle between the two lines.

If you have three points A B C and you want to find the angle ABC, then you can get that from the dot product of vector BA with vector BC. That may be easier for you.

 

[CraigH]

Ah okay I see what you mean, in my case $S$ is the origin. So I am looking for the angle between the line $S E$ and the vertical. I'll edit the original post and clarify this.

 

[Simon Bridge]

Please don't edit the original post, it makes the resulting conversation into nonsense and makes it useless for helping others with the same problem as you.
You have supplied with two ways to find the angle between the line SE and the k unit vector.