mathematics - How do I interpret the dot product of non-normalized vectors?

I know that if you take the dot-product of two normalized vectors, you get the cosine of the angle between them.

But if I take the dot-product of two non-normalized vectors (or one normalized, one not), how can I interpret the resulting scalar?

Answer

Others have pointed out how you can use the sign of the dot product to broadly determine the angle between two arbitrary vectors (positive: < 90, zero: = 90, negative: > 90), but there's another useful geometric interpretation if at least one of the vectors is of length 1.

If you have one unit vector U and one arbitrary vector V, you can interpret the dot product as the length of the projection of V onto U:

Equivalently, (U · V) is the length of the component of V pointing in the direction of U. ie. You can break V into a sum of two perpendicular vectors, V = (U · V)U + P, where P is some vector perpendicular to U.

This is helpful for rewriting a vector from one coordinate system in terms of a different basis, or for removing/reflecting the component of a vector that's parallel to a particular direction while keeping the perpendicular component intact. (eg. zeroing the component of a velocity that would take an object through a barrier, but allowing it to slide along that barrier, or rebounding it away)

I'm not aware of a convenient geometric interpretation of the dot product when both vectors are of arbitrary length (other than using the sign to categorize the angle).