A good answer might be:

Say that two vectors s and t have a dot product that is zero.

Pure Orthogonality Detector

When two vectors are orthogonal (to each other) then their dot product is zero, regardless of their lengths. The dot product "detects" orthogonality no matter what the lengths.

Now look at the dot product of a vector with itself:

v · v = |v||v|cos 0° = |v||v| 1.0 = |v|2

The dot product of a vector with itself yields the square of its length, or:

|v|   =   (v · v)

Used in this fashion, the dot product is a pure length detector. Since the two properties of a vector are length and orientation, you might suspect that the dot product is useful.