Calculate angle between two vectors, given their rotation w.r.t. a third vector. I have three vecto

Kolten Conrad 2022-07-08 Answered
Calculate angle between two vectors, given their rotation w.r.t. a third vector.
I have three vectors, a , b , and c in n-dimensional space. I know the coordinates of all three vectors and their dot products. Both a and b are rotated away from c by an angle α, in their own respective directions, obtaining a and b . What is the angle between a and b ?
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Answers (1)

sniokd
Answered 2022-07-09 Author has 22 answers
Step 1
Assuming that a,b,c have norm 1, we have
a = 2 ( a c ) a c b = 2 ( b c ) b c
Indeed, this implies a a = a c and a c = 2 ( a c ) 2 1 which is the cosine of the double angle. Alternatively, it is obvious geometrically that a + c 2 is the orthogonal projection of c on a, that is to say ( a c ) a
Hence
a b = 4 ( a c ) ( b c ) ( a b ) 2 ( a c ) 2 2 ( b c ) 2 + 1

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