I have a circle (in attachment)
and I need to prove that it is a circle but not some kind of ellipse
by proving 1. vector 1.vector1=vector2.vecto2 and 2. vector1.vector2=0.

I know how to prove the first one, but I have no idea how to prove the second one (I want to ask that how the dot product is zero relate to whether it is a circle?)

When the dot product of the two vectors equal to zero, it means that the two vectors are perpendicular to each other. If they are perpendicular to each other, you will be comparing two axes of the ellipse (an ellipse can be a circle, and I'm using this term to show that I'm not making any assumptions), just like you would compare the major and minor axes in an ellipse. If the magnitudes of both the minor and major axis is the same, then it's a circle (provided by the first part). Now, you have to prove that you did indeed compare minor and major axes, otherwise, it wouldn't be true.