What does it mean if the angle between two vectors is 180 degrees? Does it simply imply that they're quite different, or is there a special implication?
When you take the inner product between two vectors that are 180 degrees apart, the result has a negative sign and large magnitude to indicate that the vectors line up quite poorly - they have completely opposite directions!
…Or perhaps one might say that they're very similar, and differ only in a superficial way! (By a sign). Just depends on what you're doing.
In that sense professor, Will the vectors of image and its negative be at an angle of 180 degrees?
That would make sense. Every pixel in the negative would be the opposite of the corresponding pixel in the original photo, including properties such as brightness, color, etc. It follows that no matter how we define the vector for an image, its negative would be exactly opposite and thus produce a vector exactly opposite the original.