通过使用向量作为参考点来计算向量之间的角度：

Question

I have been trying to find a fast algorithm of calculating all the angle between n vectors that are of length x. For example if x=3 and n=4, my data would look something like this:

A: [1,2,3]
B: [2,3,4]
C: [...]
D: [...]

I was wondering is it acceptable to find the the angle between all of be vectors (A,B,C,D) with respect to some fix vector (ie X:[100,100,100,100]) and then the subtract the angles of (A,B,C,D) found with respect to that fixed value, to find the angle between all of them. I want to do this because I would only have to compute the angle once and then I can subtract angles all of my vectors to find the different between them. In short, I want to know is it safe to make this assumption?

angle_between(A,B) == angle_between(A,X) - angle_between(B,X)

and the angle_between function is the Cosine similarity.

Answer 1

That approach will only work for 2-D vectors. For higher dimensions any two vectors will define a hyperplane, and only if the third (reference) vector also lies within this hyperplane will your approach work. Unfortunately instead of only calculating n angles and subtracting, in order to determine the angles between each pair of vectors you would have to calculate all n choose 2 of them.