3 ms·
It is for unit vectors.
by marginalia_nu 3y ago
It is for unit vectors.
- Solvency 3y agoMy math is weak. I thought dot product measures alignment. Help me grok what "distance" means here?
- mecsred 3y agoUnit vectors have a normalized length, so the only difference between two unit vectors is the "heading". The angle between them can be interpreted as the the "difference" or distance, between them. Since you can compose any vector in the space out of unit vectors, you can extend the concept. See above comment inner product -> metric space.
- marginalia_nu 3y agoHmm, I think I got my math wrong too. What is a proper distance function is (1 - |a*b|). That's proportional to the distance between those points on the plane spanned along the circumference of the circle/sphere/hypersphere as it's projected onto that plane.
- sfpotter 3y agoThe dot product of two unit vectors is the cosine of the angle between them. Since cos^-1 (u*v) is the angle between them, or the geodesic distance between u and v regarded as points on the sphere, which is a distance. But the dot product is not. If u = v, then u*v = 1, but for two points not equal to zero, the distance between them should always be zero. Remember that u*v = 0 if u and v are perpendicular. If neither u nor v are 0, then for them to be perpendicular, they must certainly not be equal. The dot product does not behave like a distance.