Thoughts about the rate of change of the dot product while the vectors are rotated.

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Given that the dot product is minimized when the two vectors are antiparallel and maximized when the two vectors are parallel and zero when they are orthogonal, what’s the rate of change of the dot product as the angle between the vector is varied?

I did ask him a question after class that I started thinking about: given that two vectors of equal magnitude, their dot product is maximized when they're parallel, minimized when they are antiparallel, and zero when they are orthogonal, what's the rate of change of the dot product as a function of the angle changing between them? What I then realized was that this is very related to the unit circle if you think of the x-axis as one vector and the rotating radius as another vector (they're both of equal magnitude because they're both radii). That led me to think that the rate of change of the dot product with relation to the angle between them is related to a trigonometric function.