The dot product allows us to derive two important mathematical inequalities. The proofs are left
to the textbook, but they are listed here:
1) Cauchy-Schwarz Inequality: |u • v| ≤ ||u|| ||v||
2) Triangle Inequality: ||u + v|| ≤ ||u|| + ||v||
We showed the triangle inequality previously as a note regarding vector length, but it is also an important mathematical inequality that is used elsewhere, and should be remembered. It is easy to see in the xy plane that this is true (hence the name triangle inequality),
It should be noted that in R2 and R3, the triangle inequality is only equal if the two vectors are colinear, that is, they lie on exactly the same straight line when put tip-to-tail (such as vectors a and b in the above picture.)