By definition (probably a limited one because this is an intro course to linear algebra) the zero vector is perpendicular to any vector (this is because zero vector dot any other vector gives a dot product of zero).

So, am I right to say that the zero vector is just a point in n-dimensional space?

If this is so, then is the zero vector perpendicular to the zero vector? By the definition from above it is, but if the zero vector is a point in n-dimensional space then how is one point perpendicular to another point?

Admittedly I have made the assumption that the zero vector is a point in space with no directionality. If this is not the case, then please tell me what the zero vector is (in your explanation keep in mind that I am a biochemist taking this course as an interesting elective and as such my mathematical knowledge is limited).

Thanks!